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Question 728593: Please help
Solve: (sqrt of 2x - 3) = x - 1

Solve: (sqrt of 3x - 3) - (sqrt of x) = 1

Found 2 solutions by greenestamps, MathTherapy:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

You can get good mental exercise by solving these equations using logical reasoning and simple mental arithmetic.

Start by assuming (hoping) that the solutions are integers. Then use logical reasoning to determine the solution(s).

(1)

In this equation, we want (2x-3) to be a perfect square. And if x is an integer, the (2x-3) is an odd perfect square. So look at values of x for which (2x-3) is an odd perfect square.

(a) 2x-3 = 1; 2x = 4; x = 2. sqrt(2x-3) = sqrt(1) = 1; and 2-1 = 1. So x = 2 is a solution.

(b) 2x-3 = 9; 2x = 12; x = 6. sqrt(2x-3) = 3 but x-1 = 6. x = 6 is not a solution.

For larger values of x, "x-1" will increase faster than "sqrt(2x-3)", so there are no more solutions.

ANSWER: x = 2

(2)

In this equation, we want x and (3x-3) to both be perfect squares.

(a) x = 1; 3x-3 = 0; both are perfect squares. However, trying x = 1 in the equation gives us "0-1 = 1", which is not true. Note that the two sides of the equation evaluate to numbers that are opposites. This is a common occurrence in equations like this, where the formal solution process including squaring both sides of an equation leads to extraneous solutions.

So x = 1 is NOT a solution.

(b) x = 4; 3x-3 = 9 is also a perfect square; and sqrt(3x-3) = 3 and x-1 = 3, so x = 4 is a solution.

Similarly to the first problem, for larger values of x "sqrt(3x-3)" will increase faster than "sqrt(x)", so again there are no more solutions.

ANSWER: x = 4

Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

Please help Solve: (sqrt of 2x - 3) = x - 1 Solve: (sqrt of 3x - 3) - (sqrt of x) = 1 ********************************** Again, respondent @MathLover1(20855) used the QUADRATIC EQUATION formula to solve both quadratic equations, which is UNNECESSARY, since the resultant TRINOMIALS are FACTORABLE! Furthermore, for , she got 2 values for x: x = 4, and x = 1. Obviously, when checked, the value 1 DOES NOT SATISFY this equation, thereby making 1, an EXTRANEOUS solution. So, in this case, x = 4 is the SOLE VALID and ACCEPTABLE SOLUTION.

Question 968610: Identify the min/max (vertex) value. Then sketch the graph.
f(x)-x^2-10x-30
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
Student could have gave minus sign instead of "equals" sign.

and wants vertex, and know if max or min.

Try completing-square.
;
add and subtract from, to the quadratic expression.

vertex point at (5, -55) and is a minimum.

Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

The wording of the post is certainly strange; but the intention is clear: find the vertex.

Given a quadratic function , the x-coordinate of the vertex is :

The y-coordinate of the vertex is found by evaluating the function at that value of x:

ANSWER: The vertex is (-5,-5)

The graph....

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Identify the min/max (vertex) value. Then sketch the graph.
f(x)-x^2-10x-30
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The post is written incorrectly and makes no sense.

The algorithm (the computer code) in the post by @lwsshak3 did not recognize the nonsense

and produced incorrect non-sensical gibberish, which @lwsshak3 tries to sell as the solution,

without reading the output of his code.

Question 976976: Hey ! I need help solving this.
f(x) = x^4 − x^3 − 20x^2
What are all the zeros of the function and determine the multiplicity of each zero and the number of turning points of the graph of the function.
Thank you!
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.

The answer in the post by @lwsshak3 is not precisely correct/accurate.

He correctly determined the zeros: they are 0, -4 and 5.

But when @lwsshak3 says that the multiplicity is 1 (one) for all three zeros,
this is wrong.

Multiplicity is 2 (two) for the root 0 and 1 (one) for both roots -4 and 5.

Question 1003368: Use the method of completing the square to find the standard form of the quadratic function.
f(x) = x^2 + 6x − 1
y=
x=
(x,y)=
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
In case you are looking for something else, you could use to complete the square.

so function is .
This is not "Standard Form".

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Use the method of completing the square to find the standard form of the quadratic function.
f(x) = x^2 + 6x − 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The problem's formulation in the post is absolutely illiterate from the Math point of view,

since the given function is JUST WRITTEN in the standard form.

Question 299568: The product of 7 and the square of a number is 112. Find the number.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
The product of 7 and the square of a number is 112. Find the number.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This problem has two possible answers,
        while @mananth provides only one.

let the number be x
square will be x^2
7x^2 = 112
x^2 = 112/7
x^2 =16
x = 4 or -4.

Question 283230: Factor completely.
-3x^3 - 6x^2 + 189x
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Factor completely.
-3x^3 - 6x^2 + 189x
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The problem asks to factor completely.
        Instead, @mananth solved an equation, and got an incomplete solution.
        Thus, the requested assignment was not done in accordance with the request.

        I came to provide a correct solution.

-3x^3 - 6x^2 + 189x = = -3x*(x^2 + 2x - 189) = = -3x*(x+9)*(x-7). ANSWER

Solved.

Question 282341: A group plans to share the cost equally for a $200,000 plane,the group wants to find 5 more people to join ,so that the cost will decrease $2,000 per person .How people are currently in the group.

Found 3 solutions by josgarithmetic, n2, ikleyn:
Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!

PRICE PEOPLE COST 200000/n n 200000 200000/(n+5) n+5 200000 DIFF. 2000

Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
A group plans to share the cost equally for a $200,000 plane, the group wants to find 5 more people to join,
so that the cost will decrease $2,000 per person. How people are currently in the group ?
~~~~~~~~~~~~~~~~~~~~~~~

The original scenario is dollars per person. The other scenario is dollars per person. The setup equation is - = 2000. Divide both side by 1000 to make writing easier - = 2. Reduce it to the quadratic equation x^2 + 5x - 5000 = 0. Factor left side (x+25)*(x-20) = 0. The roots are x= -25 and x= 20. Since we look for a price, we accept the positive root and reject the negative one. ANSWER. Currently/originally the group consists of 20 people.

Solved.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A group plans to share the cost equally for a $200,000 plane, the group wants to find 5 more people to join,
so that the cost will decrease $2,000 per person. How people are currently in the group ?
~~~~~~~~~~~~~~~~~~~~~~

        The solution by @mananth is incorrect due to arithmetic errors inside of it.
        I came to bring a correct solution.

The original scenario is dollars per person. The other scenario is dollars per person. The setup equation is - = 2000. Divide both side by 1000 to make writing easier - = 2. Reduce it to the quadratic equation x^2 + 5x - 5000 = 0. Factor left side (x+25)*(x-20) = 0. The roots are x= -25 and x= 20. Since we look for a price, we accept the positive root and reject the negative one. ANSWER. Currently/originally the group is 20 people.

Solved correctly.

Question 275411: In a chemistry class, 6 liters of 4% silver iodide solution must be mixed with 10% solution to get a 6% solution. How many liters of 10% solution are needed?
Found 5 solutions by math_tutor2020, greenestamps, josgarithmetic, n2, ikleyn:
Answer by math_tutor2020(3838)   (Show Source):
You can put this solution on YOUR website!

Tutor mananth made an error when writing 0.04x as it should be 0.04*6 instead.
Solving 0.04*6+0.1x = 0.06*(x+6) leads to x = 3 as tutor ikleyn has shown, and as I show below.

Tutor josgarithmetic made an error on the last step. It should be 6*(0.06-0.04)/(0.1-0.06)
The "1" should be "0.1" instead
That expression evaluates to 3.
I think it's beneficial to simplify along the way to avoid a sea of numbers.

--------------------------------------------------------------------------

Here's how I would solve the problem.

x = amount of the 10% solution in liters

We have 6 liters of the 4% solution.
That contributes 0.04*6 = 0.24 liters of pure silver so far.

We add x liters of the 10% solution.
So we add 0.1x liters of pure silver to get 0.24+0.1x liters of pure silver total.

This is out of 6+x liters of solution of silver and other elements.

(amount of pure silver)/(amount of solution) = 6% goal
(0.24+0.1x)/(6+x) = 0.06
0.24+0.1x = 0.06(6+x)
0.24+0.1x = 0.36+0.06x
0.1x-0.06x = 0.36-0.24
0.04x = 0.12
x = 0.12/0.04
x = 3

Answer: 3 liters

Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

The responses you have received so far all use the standard formal algebraic method for solving the problem -- writing and solving an equation which says the sum of the amounts of silver iodide in the two ingredients is equal to the amount in the mixture.

If a formal algebraic solution is needed, then that is the standard method and almost certainly the fastest formal method.

But 2-part mixture problems like this can be solved much faster using an informal method using the ratio of the amounts of the two ingredients.

Here in words is the solution to this problem using this method.

(1) The target 6% solution is "twice as close to 4% as it is to 10%" (the difference between 4% and 6% is 2%; the difference between 6% and 10% is 4%.)
(2) That means the amount of 4% silver iodide in the mixture must be twice as much as the amount of 10% silver iodide.
(3) The mixture uses 6 liters of the 4% silver iodide, so it must use 3 liters of the 10% silver iodide.

ANSWER: 3 liters

Answer by josgarithmetic(39838)   (Show Source):
You can put this solution on YOUR website!
volume total,
v, the volume of the 10% to use
pure silver in the solution mixture,

final concentration, 6%

-

compute the expression.

Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
In a chemistry class, 6 liters of 4% silver iodide solution must be mixed with 10% solution to get a 6% solution.
How many liters of 10% solution are needed?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let x be the volume (in liters) of the 10% silver iodide solution to add. Write the balance equation for solute (silver iodide) 0.04*6 + 0.1x = 0.06*(6+x). Simplify and find x 0.24 + 0.1x = 0.36 + 0.06x, 0.1x - 0.06x = 0.36 - 0.24, 0.04x = 0.12 x = 0.12/0.04 = 3. ANSWER. 3 liters of the 10% silver iodide solution should be added.

Solved correctly.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
In a chemistry class, 6 liters of 4% silver iodide solution must be mixed with 10% solution to get a 6% solution.
How many liters of 10% solution are needed?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is incorrect starting from its first line to the end.
        It is because his governing equation is written incorrectly.

        I came to bring a correct solution.

Let x be the volume (in liters) of the 10% silver iodide solution to add. Write the balance equation for solute (silver iodide) 0.04*6 + 0.1x = 0.06*(6+x). Simplify and find x 0.24 + 0.1x = 0.36 + 0.06x, 0.1x - 0.06x = 0.36 - 0.24, 0.04x = 0.12 x = 0.12/0.04 = 3. ANSWER. 3 liters of the 10% silver iodide solution should be added.

Solved correctly.

Question 268557: please help me solve this equation: x/x+1-2=3/x-3
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

please help me solve this equation: x/x+1-2=3/x-3 ************************************************* , with x(x - 3) - 2(x + 1)(x - 3) = 3(x + 1) ---- Multiplying by LCD, (x + 1)(x - 3) (x - 1)(x + 3) = 0 --- Factorizing the trinomial x - 1 = 0 OR x + 3 = 0 --- Equating each binomial to 0 x = 1 OR x = - 3 Neither of the 2 solutions is - 1 or 3, so BOTH are VALID!

Question 268825: Please help me solve this equation:
Found 2 solutions by n2, ikleyn:
Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
Please help me solve this equation: =
~~~~~~~~~~~~~~~~~~~~~~~~~~

Your starting equation is = . The domain of this equation is the set of all real numbers except of x= -1 and x= 3. We will work over the domain, assuming that x =/= -1 and x =/= 3. Multiply both sides by LCD (x+1)*(x-3) and simplify x*(x-3) - 2(x+1)*(x-3) = 3(x+1), x^2 - 3x - 2*(x^2 +x - 3x - 3) = 3x + 3, x^2 - 3x - 2x^2 - 2x + 6x + 6 = 3x + 3, -x^2 - 2x - 3 = 0, x^2 + 2x + 3 = 0, (x+3)*(x-1) = 0. The solutions to this equation are the numbers -3 and 1. They both are in the domain of the given equation, so the solution to equation (1) are x = -3 and x = 1.

Solved completely and correctly.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Please help me solve this equation: =
~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is INCORRECT.

        His transformations, which he performs to reduce the given equation to the factored quadratic equation,
        contain a lot of arithmetic errors, and his final equation is wrong.

        The answer absents in his solution. So, his presentation is a compote of mathematical symbols
        with no mathematical sense, which may lead a reader to wrong conclusion.

        Therefore, I came to bring a correct solution.

Your starting equation is = . The domain of this equation is the set of all real numbers except of x= -1 and x= 3. We will work over the domain, assuming that x =/= -1 and x =/= 3. Multiply both sides by LCD (x+1)*(x-3) and simplify x*(x-3) - 2(x+1)*(x-3) = 3(x+1), x^2 - 3x - 2*(x^2 +x - 3x - 3) = 3x + 3, x^2 - 3x - 2x^2 - 2x + 6x + 6 = 3x + 3, -x^2 - 2x - 3 = 0, x^2 + 2x + 3 = 0, (x+3)*(x-1) = 0. The solutions to this equation are the numbers -3 and 1. They both are in the domain of the given equation, so the solution to equation (1) are x = -3 and x = 1.

Solved completely and correctly.

/\/\/\/\/\/\/\/\/\/\/\/\/

Here is my general impression about the solutions by @mananth at this forum.

I just learned several months ago, that @mananth systematically uses a computer code,
which generates files with solutions.

In many cases (approximately in 10% of cases) the solutions generated by his computer code are incorrect.

But @mananth never reads and never checks what his code produces, so @mananth
does not carry any responsibility for the quality of his solutions.

Any reader should understand it - - - @mananth does not carry any responsibility for the correctness
of his solutions: this is his principial position.

Factually, he leaves this responsibility to those tutors (like me),
who check every, each and all his solutions totally.

Question 695911: Find the product of the solutions of this equation. 3x^2 - 12x + 7 = 0
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
.
Find the product of the solutions of this equation. 3x^2 - 12x + 7 = 0.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This is a nice question, and it has a nice answer.

To get the product of the roots of this equation, you do not need to solve the equation
and you do not need to find the roots explicitly, as the other tutor makes in his post.

Instead, use the Vieta's theorem.

It says that the product of the roots of a quadratic equation   ax^2 + bx + c = 0

is always   ,   or the constant term divided by the leading coefficient at x^2.

In your case, the product of the roots is   ,   so you get the answer instantly, without making long calculations.

The Vieta's theorem works for real roots and for complex roots: in other words, it works for any quadratic equation.

It is UNIVERSAL.

This problem is specially given to you to check if you are familiar with the Vieta's theorem,
and if you are not familiar with it, it is a good chance to learn it and to see how perfectly it works.

It is also a good question for an interview, since it allows to learn momentarily,
if a person knows Algebra and generally a school Math uniformly.
How a person answers this question, it demonstrates/reflects the person's mathematical culture.

Question 62776: help please
solve the rational inequality (x-2)/(x+3) is less than or equal to 0
is it: (-3,2] (-oo,-3) (-oo,-3)U{2,oo) or [2,oo)
i am very confused
Found 2 solutions by n2, ikleyn:
Answer by n2(91)   (Show Source):
You can put this solution on YOUR website!
.
solve the rational inequality (x-2)/(x+3) <= 0
~~~~~~~~~~~~~~~~~~~~~~~~

They want you solve this inequality <= 0. (1) The left side rational function can be non-positive if and only if EITHER the numerator is non-positive and denominator is positive x - 2 <= 0 and x + 3 > 0 (2) OR the numerator is non-negative and denominator is negative x - 2 >= 0 and x + 3 < 0. (3) In case (2), x <= 2 and x > -3 simultaneously, or -3 < x <= 2. In case (3), x >= 2 and x < -3 simultaneously, which has no solutions. Thus the final solution to the given inequality is this set of real numbers -3 < x <= 2, or, in the interval notation, the set (-3,2].

Solved.

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
help please
solve the rational inequality (x-2)/(x+3) is less than or equal to 0
is it: (-3,2] (-oo,-3) (-oo,-3)U{2,oo) or [2,oo)
i am very confused
~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @jai_kos in incorrect.
        It is incorrect methodologically and gives incorrect answer.
        See my correct solution below.

They want you solve this inequality <= 0. (1) The left side rational function can be non-positive if and only if EITHER the numerator is non-positive and denominator is positive x - 2 <= 0 and x + 3 > 0 (2) OR the numerator is non-negative and denominator is negative x - 2 >= 0 and x + 3 < 0. (3) In case (2), x <= 2 and x > -3 simultaneously, or -3 < x <= 2. In case (3), x >= 2 and x < -3 simultaneously, which has no solutions. Thus the final solution to the given inequality is this set of real numbers -3 < x <= 2, or, in the interval notation, the set (-3,2].

Solved.

The error made by @jai_kos is that when he multiplies both sides of the original inequality by (x+3),
he misses the case when (x+3) is negative, which requires different treatment.

This error, which jai_kos makes solving the problem, is a typical error, which beginners make
when trying to solve such inequalities,
until the more experienced teachers/tutors will explain their error and will show a right way solving.

Question 264812: what are the factors of 6r^2+3rs-18s^2
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
what are the factors of 6r^2+3rs-18s^2 ?
~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is incorrect - simply ignore it.
        See below my correct solution.

6r^2 + 3rs - 18s^2 = = 3*(2r^2 + rs - 6s^2) = = 3*(2r^2 + 4rs - 3rs - 6s^2) = = 3*(2r(r+2s) - 3s(r+2s)) = = 3*(2r-3s)(r+2s). ANSWER

Solved correctly.

Question 263772: Hi,
I have a quadratic factorization question that I don't know how to do...
It's a Year 11 Maths question:
4x^4-37x^2+9
I have to factorize that and I would show you what I've done so far but I have no idea how to do this question...
Thank you so much for your help! :)
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Hi,
I have a quadratic factorization question that I don't know how to do...
It's a Year 11 Maths question:
4x^4-37x^2+9
I have to factorize that and I would show you what I've done so far but I have no idea how to do this question...
Thank you so much for your help
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Factorize your polynomial step by step

4x^2 - 37x^2 + 9 =
= 4x^4 - 36^2*x^2 - x^2 + 9
= 4x^2(x^2-9) - (x^2-9)
= (4x^2-1)*(x^2-9)
= (2x+1)(2x-1)(x+3)(x-3)

The difference between my solution and the @mananth solution is that
@mananth makes his transformations on an equation (which is not needed),
while I make transformations over the polynomial, exactly as it is requested.

Question 314826: SOLVE BY USING QUADRATIC EQUATION
FIND THE CONSECUTIVE INTEGERS SUCH THAT THE SUM OF THEIR SQUARES IS 245
PLEASE HELP I HAD QUESTIONS LIKE THIS ON MY FINAL BUT I COULD NOT UNDER STAND HOW TO WORK THIS PROBLEM
Found 3 solutions by KMST, timofer, MathTherapy:
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website!
There are only two sets of consecutive integers with squares that add up to 245.
and
The sum of the squares of a set of less than 3 or more than 3 consecutive integers is never 245.
The sum of the squares of 15 or more consecutive integers is always more than 245.

Answer by timofer(159)   (Show Source):
You can put this solution on YOUR website!
"THE CONSECUTIVE INTEGERS"
How many consecutive integers? The question or description is not complete.

If you want THREE consecutive integers, you should have no trouble finding sum of their squares to be 8, 9, 10.

If middle is n, then the consecutive integers n-1, n, n+1.
Setup and solve .

Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

SOLVE BY USING QUADRATIC EQUATION FIND THE CONSECUTIVE INTEGERS SUCH THAT THE SUM OF THEIR SQUARES IS 245 PLEASE HELP I HAD QUESTIONS LIKE THIS ON MY FINAL BUT I COULD NOT UNDER STAND HOW TO WORK THIS PROBLEM ****************************************************************************************************** I don't know how the other person can say that these 2 integers are 11 and 12. They are NOT!! Correct answer: NO 2 integers'/consecutive integers' sum of squares is 245. So, NO SOLUTION!!

Question 25608: I am trying to help a homeschooled student with algebra. We are stumped on a problem. It's on a page where you are supposed to use the quadratic formula to solve but i'm having a hard time setting up the equation. It says; working alone Colleen can paint a house in 2 h less than James. Working together, they can paint the house in 10 h. How long would it take James to paint the house by himself? Thank you!
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

I am trying to help a homeschooled student with algebra. We are stumped on a problem. It's on a page where you are supposed to use the quadratic formula to solve but i'm having a hard time setting up the equation. It says; working alone Colleen can paint a house in 2 h less than James. Working together, they can paint the house in 10 h. How long would it take James to paint the house by himself? Thank you! *************************************** The other person whu responded, says, "So it takes Jmes 6 hrs, so then it takes Colleen; 6-2=4 =)". NO!!! It DOESN'T!! Let time it takes James, be J Then time it takes Colleen = J - 2 With both taking 10 hours to do the job, working together, we get the following equation: ----- Multiplying by LCD, 10J(J - 2) You now have your quadratic equation. I take it you know wnhat to do now, and after solving, correct? If not, it’s that you MUST CHECK your solutions to ensure that they make sense, in this particular scenario.

Question 1180203: the sum of the squares of two consecutive numbers is 61.What are those numbers
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
the sum of the squares of two consecutive numbers is 61. What are those numbers
~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Two other tutors provided a formal algebra solution.
        Which is good to develop your technique.
        But I will give here simple MENTAL solution - - - which is good to develop your mind.

The sum of the squares of two consecutive integer numbers is 61. So, we can expect that each of the two squares is about 30 (about half of 61). Such two squares of integers closest to 30 are 25 and 36 - so check 25 + 36 = 61 <<<---=== correct ! Thus the solution is the pair (5,6). Recalling the signs, we detect another pair (-6,-5), too.

Solved MENTALLY.

Many other similar problems can be solved similarly.
It is a good technique to use it in competitions for fastest solution.

Question 76792: Simplify.
I am having a really hard time with this problem and need some help please thanks.
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

Simplify. I am having a really hard time with this problem and need some help please thanks. ********************************************************************************* Why would you? This is so EASY!! = = = = OR = =

Question 1177833: A landscape architect is planning a circular flower
bed that will be surrounded by a ring of paving
stones, 25 cm wide. The area of the ring of paving
stones will be half the area of the flower bed.
Determine the radius of the flower bed, to the
nearest centimetre.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
A landscape architect is planning a circular flower bed that will be surrounded by a ring of paving
stones, 25 cm wide. The area of the ring of paving stones will be half the area of the flower bed.
Determine the radius of the flower bed, to the nearest centimetre.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution by @mananth is incorrect due to arithmetic errors on the way.
        I came to bring a correct solution.

pi*(r+25)^2 - pi*r^2 = (1/2)*pi*r^2 Cancel pi in both sides (r+25)^2 - r^2 = (1/2)*r^2 r^2 + 50r + 625 - r^2 = (1/2)*r^2 50r + 625 = (1/2)r^2 r^2 - 100r - 1250 = 0 Use the quadratic formula and find r = . It gives a unique meaningful positive solution x = = 111 cm, rounded. ANSWER

Solved.

Question 33877: sqrt(2x-5)-sqrt(x-3)=1
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

sqrt(2x-5)-sqrt(x-3)=1 ====================== Whatever the other person who responded did, doesn't make sense, at all, to this author. Yet, it's quite surprising that he/she got one of the solutions. But, there's another one! , with ----- Adding to both sides ---- Squaring each side (x - 7)(x - 3) = 0 x - 7 = 0 OR x - 3 = 0 x = 7 OR x = 3 As 7 > 3 and 3 = 3, the above constraint, is satisfied. Therefore, both solutions are ACCEPTABLE!!

Question 107371: I need help with this kind of problem:
Find the vertex, the y-intercept, and symmetric point, and use these to sketch a graph.
y=2x^2 -11x -12
I am having problems solving for the vertex.
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

I need help with this kind of problem: Find the vertex, the y-intercept, and symmetric point, and use these to sketch a graph. y=2x^2 -11x -12 I am having problems solving for the vertex. ============================================ Formula for the x-coordinate of the vertex of a parabola: . Compare to: In this case, b = - 11, and a = 2. So, the x-coordinate of the vertex of THIS parabola is: Now, substitute the x-value into the ORIGINAL equation to get the y-coordinate of the vertex, as follows: <==== y-coordinate of the vertex of THIS parabola Vertex of parabolic equation, :

Question 1179148: the distance between London and New York is 320 Km. A train takes x hours to travel between London and New York.
(a) Write down an expression in terms of x, for the average speed of the train [1]
(b) A car takes 2 ½ hours longer than a train to travel between London and New York. The average speed of the train is 80 Km/h greater than the average speed of the car. Form an equation in x and show that it simplifies to

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
the distance between London and New York is 320 Km. A train takes x hours to travel between London and New York.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When I read this, my hair stands on end and my eyes go square.

This writer needs to learn from a geography textbook and to pass a school exam on Geography.

This writing is a style of @CPhill.

Such gibberish should not be spread in the Internet.

Question 96205: Please help me solve this problem:
y^4/3 = -3y
This is what I have done:
y^4/3 + 3y =0
1. The exp of y is understood to be 1...so does this mean that I need to find the commoon denominator of the exp so that I could add? Should the exp of 3y then be 1/3?
y^4/3 + 3y^1/3= 0
2. Then in the book it says to factor out maybe 1/3?
This is where I get confused, because the problem in the book that is simmilar looks like this:
x^3/2= x^1/2
x^3/2- x^1/2= 0
x^1/2(x-1)) =0 How are they factoring out x^1/2? Is x^1/2(x-1) = x^3/2 -x^1/2
3. So once they get x^1/2= 0 and x-1=0
so the answer is x=0 and x=1
Found 2 solutions by greenestamps, MathTherapy:
Answer by greenestamps(13367)   (Show Source):
You can put this solution on YOUR website!

...

This is what I have done:
y^4/3 + 3y =0
1. The exp of y is understood to be 1...so does this mean that I need to find the commoon denominator of the exp so that I could add? Should the exp of 3y then be 1/3?
y^4/3 + 3y^1/3= 0

No.

As the other tutor says, you seem to be trying to make this problem look like the example in your book, but it is very different.

In the expression "3y" in this problem, the understood exponent "1" is only the exponent of "y" -- it is not the exponent of "3y".

Leave the equation as

y^4/3 + 3y =0

Then, as the other tutor shows, factor out the common factor "y" on the left and solve the problem from there.

Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

Please help me solve this problem: y^4/3 = -3y This is what I have done: y^4/3 + 3y =0 1. The exp of y is understood to be 1...so does this mean that I need to find the commoon denominator of the exp so that I could add? Should the exp of 3y then be 1/3? y^4/3 + 3y^1/3= 0 2. Then in the book it says to factor out maybe 1/3? This is where I get confused, because the problem in the book that is simmilar looks like this: x^3/2= x^1/2 x^3/2- x^1/2= 0 x^1/2(x-1)) =0 How are they factoring out x^1/2? Is x^1/2(x-1) = x^3/2 -x^1/2 3. So once they get x^1/2= 0 and x-1=0 so the answer is x=0 and x=1 I think you're CONFUSING yourself, and probably mixing up this given problem with the one in your book. Anyway, I hope you can follow what I've presented below. ---- Adding to both sides ---- Factoring out y OR y = 0 --- Cubing each side

Question 715207: why would 4 plus/minus the square root of 32 divided by 2 equal 2 plus or minus the square root of 8 and not be 2 plus or minus the square root of 16 ?
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

why would 4 plus/minus the square root of 32 divided by 2 equal 2 plus or minus the square root of 8 and not be 2 plus or minus the square root of 16 ? This obviously is the calculation of the roots, using this, the quadratic equation formula. You gave: = = = = = = , or

Question 1037220: how get sqrt(3+sqrt(5-sqrt(13+sqrt(48)))) show than (sqrt(2)+sqrt(6))/2
helpme please

Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

how get sqrt(3+sqrt(5-sqrt(13+sqrt(48)))) show than (sqrt(2)+sqrt(6))/2 helpme please SIMPLIFYING SIMPLIFYING SIMPLIFYING Proof that sqrt(3+sqrt(5-sqrt(13+sqrt(48)))) = (sqrt(2)+sqrt(6))/2, COMPLETED!

Question 449465: Solve the equation 25x2 - 64 = 0 using the quadratic formula. (Notice b = 0. Enter solutions from smallest to largest.)
x = ?
x = ?
Please show me the steps to get the steps.
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Solve the equation 25x2 - 64 = 0 using the quadratic formula. (Notice b = 0. Enter solutions from smallest to largest.)
x = ?
x = ?
Please show me the steps to get the steps.
~~~~~~~~~~~~~~~~~~~~~~~~~

It's ridiculous to solve this quadratic equation using the quadratic formula.

The instruction is too stupid.

Question 442619: Hi,
I have an algebra question that states the following: (not sure where to go with this)
If h = -16t^2 + 48t represents the height of a rocket, in feet, t seconds after it was fired, when will the rocket hit the ground?
It does give a hint: "The rocket is on the ground when h = 0"
So it states that the rocket will hit the ground after _______ second(s).

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
Hi,
I have an algebra question that states the following: (not sure where to go with this)
If h = -16t^2 + 48t represents the height of a rocket, in feet, t seconds after it was fired, when will the rocket hit the ground?
It does give a hint: "The rocket is on the ground when h = 0"
So it states that the rocket will hit the ground after _______ second(s).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is interesting to know how physicists solve such problems. The problem says that the model rocket was launched from the ground level with vertical speed of v = 48 feet per second, and the gravity acceleration is g = 32 ft/s^2. Then according to kinematic, the time to get the highest point is t = = 1.5 seconds, and the same time of 1.5 seconds is needed for the model rocket to fall from the highest point to the ground. The sum 1.5 + 1.5 = 3 seconds is the time from the starting moment to the moment when the model rocket hits the ground.

Solved.

Question 1210513: Scientists launched a rocket which can be modeled by the equation h = -16t2 + 160t + 1, where t is time in seconds, and h is the distance from the ground in feet.
a) Find the maximum height of this rocket.
____ feet

b) When does the maximum height occur?
_ seconds

Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website!
This problem can be solved by finding the vertex of the parabola described by the height function:
$$h(t) = -16t^2 + 260t + 1$$
This is a quadratic equation in the form $h(t) = at^2 + bt + c$, where $a = -16$, $b = 260$, and $c = 1$. Since $a$ is negative, the parabola opens downward, and the vertex represents the maximum height.
### a) Find the maximum height of this rocket.
The maximum height occurs at the time $t$ given by the axis of symmetry formula:
$$t = -\frac{b}{2a}$$
The maximum height ($h_{\text{max}}$) is found by substituting this time back into the height equation $h(t)$.
#### Step 1: Find the time of maximum height ($t$)
$$t = -\frac{260}{2(-16)} = -\frac{260}{-32} = \frac{260}{32}$$
$$t = 8.125 \text{ seconds}$$
#### Step 2: Calculate the maximum height ($h$)
Substitute $t = 8.125$ into the height equation:
$$h_{\text{max}} = -16(8.125)^2 + 260(8.125) + 1$$
I will use the code interpreter to perform the calculation precisely.
```python?code_reference&code_event_index=2
# Given function parameters
a = -16
b = 260
c = 1
# Calculate time (t) of maximum height (t = -b / 2a)
t_max = -b / (2 * a)
# Calculate the maximum height (h) by substituting t_max into the equation h = at^2 + bt + c
h_max = a * (t_max**2) + b * t_max + c
print(f"Time of maximum height (t_max): {t_max}")
print(f"Maximum height (h_max): {h_max}")
```
```text?code_stdout&code_event_index=2
Time of maximum height (t_max): 8.125
Maximum height (h_max): 1057.25
```
### a) Find the maximum height of this rocket.
Substituting $t = 8.125$ into the equation:
$$h_{\text{max}} = -16(8.125)^2 + 260(8.125) + 1$$
$$\mathbf{h_{\text{max}} = 1057.25 \text{ feet}}$$
### b) When does the maximum height occur?
The maximum height occurs at the time calculated by the axis of symmetry formula:
$$t = -\frac{b}{2a} = 8.125$$
$$\mathbf{t = 8.125 \text{ seconds}}$$
-----
**Final Answers:**
a) Find the maximum height of this rocket.
**1057.25** feet
b) When does the maximum height occur?
**8.125** seconds

Question 1146014: 1. (a) resolve into partial fractions 2x/[(x-2)(x+5)]
(b) resolve into partial fractions 1/(x^2-x)
Answer by MathTherapy(10858)   (Show Source):
You can put this solution on YOUR website!

1. (a) resolve into partial fractions 2x/[(x-2)(x+5)] (b) resolve into partial fractions 1/(x^2-x) 1. (a) resolve into partial fractions The other person’s solution: A = , and B = , is WRONG!! = ---- Multiplying by LCD, (x - 2)(x + 5) 2x = A(x + 5) + B(x - 2) ---- Equating NUMERATORS, since denominators are the same 2(2) = A(2 + 5) + B(2 - 2) ---- Substituting 2 for x to determine the value of A 4 = 7A 2(- 5) = A(- 5 + 5) + B(- 5 - 2) ---- Substituting - 5 for x to determine the value of B - 10 = - 7B (A, B) = (, ) Therefore, = = 1. (b) resolve into partial fractions Use the same concept above, to decompose this PROPER FRACTION too. Before doing so though, we FACTORIZE the denominator in to get: , and then: =

Question 732314: If the positive integer x leaves a remainder of 2 when divided by 8, what will the remainder be when x + 9 is divided by 8?
Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
If the positive integer x leaves a remainder of 2 when divided by 8,
what will the remainder be when x + 9 is divided by 8?
~~~~~~~~~~~~~~~~~~~~~~~

We are given that x = 8n + 2 <<<---=== "the positive integer x leaves a remainder of 2 when divided by 8" It implies x + 9 = (8n + 2) + 9 x + 9 = 8n + 11, x + 9 = 8(n+1) + 3. This means that the remainder of (x+9) when divided by 8 will be 3. <<<---=== ANSWER

Solved.