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Think about this! 70% acid solution is 30% water and the 50% acid solution is 50% water.
You want to add x liters of water to the 2.5 liters of 30% water/acid solution to make (2.5 + x) liters of 50% water/acid solution. Change the percents to their decimal equivalents 30% = 0.3, 50% = 0.5 Now you can set up your equation to solve the problem.
Simplify and solve for x.
Subtract 0.5x from both sides.
Subtract 0.75 from both sides.
Divide both sides by 0.5

The Chemist will need to add 1 liter of water to the 2.5 liters of 70% acid solution to obtain 3.5 liters of 50% acid solution.

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Use similar triangles to solve this problem.
The yard stick (1 yd = 3 ft.) is the vertical leg of one right triangle and its shadow (2 ft) is the base.
The tree (h ft.) is the vertical leg of the second right triangle and its shadow (30 ft) is the base.
Recalling that "corresponding sides of similar triangles are proportional", you can form a proportion to find h, the height of the tree.
Multiply both sides by 30.
= 45 feet.

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Try this:
Let the first number be x, then the next consecutive number will be x+1 and the third consecutive number will be x+2 and they should all add up to 600.
Simplify.
Subtract 3 from both sides.
Divide both sides by 3.

The three consecutive numbers are: 199, 200, and 201.

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HI Wyatt, here's a formula that could help solve your problem. If you're interested, it came out of a math tables book (CRC Standard Mathematical Tables) which I have used since the early fifties in college.

Since you know C (the chord length) and h (the length of that part of the radius between the chord and the circumference), you can, with a little algebraic manipulation, solve for R, the radius.
Square both sides.
Divide both sides by 4h.
Add h to both sides.
Finally, divide both sides by 2.
This can be simplified a bit.

Now let's plug in your values of C (30") and h (2.5") and grind away.



inches.

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Ok, let's see if we can unravel the loop that you've ben thrown for.
First, you will need to learn to recognise cubic terms.
and
Rewrite your expression:

Once you see this, then you'll that your polynomial expression is the "difference of two cubes" for which there is a factored form, which, in general, looks like:
Thus, with no further ado, we can proceed from here.
Simplify.

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The number of triangles formed by drawing diagonals from one vertex to every other vertex in a polygon is always 2 less than the number of sides of the polygon. Number of triangles = n - 2 where n is the number of sides in the polygon.

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Well, you started out ok, but when you got to the point of removing the second set of parentheses, you apparently forgot that it was preceeded by a negative sign which affects the sign of every term within these parentheses. So when you take the parentheses away, you must change the sign of each term within.
Solve for x:
Perform the indicated multiplications.
Simplify. Remove the parentheses.
Collect like-terms.
Subtract 3 from both sides.
Use the quadratic formula to solve for x.



The discriminant is positive (+232) so the solution has two real roots.
The roots are:

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The equation: describes a parabola. You need to find the location of the vertex of this parabola which will be a maximum since the parabola opens downwards (-16t^2). The horizontal coordinate (t-component) of the vertex is given by: . This corresponds to the time, t, at which the firework will reach its maximum height.
So, the maximum height will occur at: = secs = 6.25 seconds. This is when the firework should be detonated for it to explode at the greatest height.
To find this maximum height, substitute t = 6.25 into the original equation and solve for the height, h. It's easier to use the fractional form, 25/4



feet.

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Solve:
Multiply both sides by x
Add x to both sides.
Subtract 1 from both sides.
Solve this quadratic equation using the quadratic formula:


x = + or -

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Solve for x:
Cross-multiply.
Perform the indicated multiplication on both sides.
Subtract x^2 from both sides.
Add 2x to both sides.
Subtract 1 from both sides.
Factor an x on the right side.
App;y the zero products principle.
and/or
If , then
The two roots are:
x = 0
x = -5
Finally:
The smaller number is -5
The larger number is 0

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As you probably know, the base angles in an isosceles triangle are equal. Let's call one of these base angles x.
And you are also most likely aware that the sum of the interior angles of any plane triangle is 180 degrees. From the problem description, we can write: (2x - 40) is the vertex angle.
Putting it all together then, we can wite:
Simplify and solve for x.
Add 40 to both sides.
Divide both sides by 4.

Each base angle is 55 degrees.
The vertex angle is: 2(55) - 40 = 110 - 40 = 70 degrees.
Check: 55 + 55 + 70 = 180

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Solve:
This can be factored by treating the x^4 as (x^2)^2.

Apply the zero products principle.
and/or
If , then and or
If , then and or
The four roots are:
x = -1
x = 1
x = -3
x = 3
So, in increasing order, we have:
x1 = -3
x2 = -1
x3 = 1
x4 = 3

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What are the factors of 18x^2?
(x)(18x) or (-x)(-18x) = 18x^2
(2x)(9x) or (-2x)(-9x) = 18x^2
(3x)(6x) or (-3x)(-6x) = 18x^2
What are the factors of -5?
(1)(-5)
(-1)(5)
For problem 6) you need the right combination that will make -27x for the middle term.
Try (3x - 5)(6x + 1) and check using FOIL:
=
For problem 8) you need the right combination that will make +9x (or +9y) for the middle term. Use y instead of x from the above factors.
Try (3y - 1)(6y + 5) and check using FOIL.
=

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Here's one approach: the 30 gallons of 70% antifreeze solution is 30% water and you want to make a solution that is 60% antifreeze and 40% water. So you will add x gallons of water to the 30% water solution.
Change the percents to their decimal equivalents.
Simplify and solve for x, the number of gallons of water to be added.
Subtract 0.4x from both sides.
Subtract 9 from both sides.
Finally, divide both sides by 0.6

You will need to add 5 gallons of water to the 30 gallons of 70% antifreeze solution to make 35 gallons of 60% antifreeze solution.
Check:

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You want to get your equation into the standard form for a circle with center at (h, k) and radius r. It will look like this:

To do this, you must:
1) Complete the square in the x-terms.
2) Complete the square in the y-terms.
3) Factor the resulting x-trinomial.
4) Factor the resulting y-trinomial.
5) Simplify, if necessary, into the standard form.
Complete the squares by adding the square of half the coefficient of x/y to both sides.
Factor the x and the y trinomials and simplify the right side of the equation.
Compare this with the standard form.
The center of the circle is at (-3, 5/2) and the radius is

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Let's start by finding the hourly rates at which Dave and Dave & Howard work.
If Dave can do the entire job in 8 hours, then he can do 1/8 of the job in 1 hour.
After dave has worked for 1 hour, he has completed 1/8 of the job, leaving 7/8 to be finished by Dave & Howard in 2 hours.
So, if Dave & Howard can do 7/8 of the job in 2 hours, then they can do 7/16 (just half of 7/8) of the job in 1 hour.
Now we'll subtract Dave's rate from Dave & Howard's combined rate leaving us with Howard's hourly rate.
H&D - D = H
7/16 - 1/8 or 7/16 - 2/16 = 5/16
Howard can do 5/16 of the job in 1 hour, therefore, it will take him 16/5 hours to do the entire job alone.
16/5 hours = 3 1/5 hours = 3 hours and 12 minutes.

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You need to set up your equation from the problem description as follows:
$x at 8% + $(1000-x) at 10% = $86 Now you solve for x, the amount in the first bank. First change the percents to their decimal equivalents. 8% = 0.08, 10% = 0.1 Now write the algebraic equation.
Simplify and solve for x.

Subtract 100 from both sides.
Divide both sides by -0.02

Answer:
$700.00 was deposited in the first bank at 8% interest per annum.
$300.00 was deposited in the second bank at 10% interest per annum.
Check:
0.08($700.00) + 0.1($300.00) = $56.00 + $30.00 = $86.00

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Let the side of the original square of cardboard be x cm.
If you now cut out 3 cm squares from each corner, the remaining side of the cardboard square is now (x - 2*3) = (x - 6) cm and this is the size of one side the bottom of the finished box.
The dimensions of the finished box is then, (x - 6)cm by (x - 6)cm by 3cm high.
The volume of the finished box is given as 60 cm^3. So now we can set up the equation to solve for x, the length of the side of the original square from which the box is made. We'll use the formula for the volume of a rectangular prism (AKA box): V = Bh where B is the area of the base and h is the height of the box.
Expand the left side.
Divide both sides by 3.
Subtract 20 from both sides.
Solve this quadratic equation using the quadratic formula:



or
Let's look at the first root: = = 10.47
Now the second root: = = 1.53
Clearly, the second root can be discarded because if that were the length of the side of the original square, then you could never have cut two 3-cm corners from it.
The first root: x = 10.47cm (Approx) is the length of the side of the original square.
Check:
This is not quite 60 because when we took the square root of 5, we got an approximate value.

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The measure of the circumference of a circle (also a wheel) is given by:
or, if you know the radius, since D = 2r.
You want to find the radius, so use:
and solve this for r. Divide both sides by
Substitute 42" for C and 3.14 for

Round to nearest 1/100th
r = 6.69 inches.
Each spoke is 6.69 inches

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The sum, S, of the measures of the interior angles of any polygon is given by:
Where: n is the number of sides in the polygon.
For a hexagon, which has 6 sides, you would have:

Is this enough to get you going?

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You'll need to set up the equation for the following:
8 oz of 30% syrup mixture + 4 oz of x% syrup mixture = 12 oz of 42% syrup mixture. Change the percents to decimals. The equation is:
Simplify and solve for x.
Subtract 2.4 from both sides.
Divide both sides by 4.
Convert to a percent.
x = 66%

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It's no surprise you are lost when you try integers as the cube root of 81.
You will not find an integer that, when cubed, will give you 81.
The cube root of 81 is an irrational number = 4.32674871092...
You could simplify it to:
or

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Solve for x by completing the square.
Complete the square in the x-terms by adding the square of half the coefficient of x to both sides. This means adding .
Factor the left side.
Take the square root of both sides.
Subtract 3/2 from both sides.
or
or

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Let the positive number be n.
Square both sides.
Multiply both sides by 1600.
Divide both sides by n.

The positive number is 1600.
Check:

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Solve by completing the square:
Complete the square in the x-terms by adding the square of half the coefficient of the x-term to both sides. This means adding to both sides.
Factor the left side.
Take the square root of both sides.
Subtract from both sides.
Simplify.

and/or

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Well, you've got the right idea but...
Solve for l:
Subtract b from both sides.
Here's where you erred. You had b - s rather than s - b
Now you would divide both sides by
You can simplify this.

Finally:

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Solve the system of equations:
1)
2) Simplify this equation.
Add the fractions of the left side.
Multiply both sides by xy
But so:
or
Now you can solve this system of two equations:
1) and
2)
Rewrite equation 1) as:
1) and substitute into equation 2)
Now solve for y.
Add 2 to both sides and simplify.
Solve by factoring.
Apply the zero products principle.
or
and/or
Now solve for x.
For y = 1, x = -y-1 = -1-1 = -2
For y = -2, x = -(-2)-1 = 2 - 1 = 1
The solutions are:
1) (-2, 1)
2) (1, -2)
Check the graph to see what this looks llike:

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Factoring trinomials can entail a lot of educated guess-and-check work.
Factor:
What are the possible factors of the term?
1) (15x)(x) or (-15x)(-x)
2) (5x)(3x) or (-5x)(-3x)
What are the possible factors of the constant term?
1) (1)(-2) or (-1)(2)
What combintation will give the x-term coefficient of -1?
1) (3)(-2) + (5)(1) This looks promising!
2) (-3)(2) + (5)(1) This won't do because you need to get a -2 as a constant term.
So, try:
Check it out using FOIL.
=

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The first step is to translate the problem from a word description to an algebraic equation: Let n be the missing number.
Simplify.
Add 2n to both sides.
Subtract 7 from both sides.
Divide both sides by 6.

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Solve the system of linear equations:

Subtract the first equation from the second equation, eliminating the x term and resulting in:
Divide both sides by 6.
Now substitute this value of y into either of the two original equations and solve for x. Let's use the first equation.
Add 4 to both sides.

Solution is: (6, 1)
You can check this solution by substituting the x and y (6, 1) of the solution into the two equations and see that they are valid.
I'll leave this as an exercise for you.

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Start with the formula for the measure of an interior angle, A, of a regular polygon of n sides:

Multiply both sides by n.
Simplify the right sides.
Add 360 to both sides.
Subtract 168n from both sides.
Divide both sides by 12.

The regular polygon has 30 sides.