Questions on Algebra: Polynomials, rational expressions and equations answered by real tutors!

I am having a problem factoring this one 14a^(2)-45a-14


Use the "ac" method
 ---- Replacing - 45a with - 49a + 4a

Find the roots of the given polynomial using the quadratic formula

    d = b^2 - 4ac = 45^2 - 4*14*(-14) = 2809

     =  =  =  =  = ;

     =  =  =  =  = .


Therefore, the decomposition of our polynomial is

    14a^2 - 45a - 14 =  = (2a-7)*(7a+2).


ANSWER.  14a^2 - 45a - 14 = (2a-7)*(7a+2).

(343/y^9)^-1/3 Write the expression using radical notation. Then simplify if possible.

The other person's answer, √7/y^3, is WRONG!!

 =  =  = 

     The domain of this equation is the set of all real numbers except of  y = -3, 
     because  (y+3)  is in the denominator of the formulas.

    Since the root y = -3 is out of the domain, the only solution to the given equation is y = 4.

It is about finding the present value of the compounded account.


The present value is   = 6102.71  (rounded to the closest cent).    ANSWER

       -   --->  write with the common denominator 14d  --->

    =  -   --->  work with this common denominator  ---> 

    =  =   --->  cancel common factor 7  ---> 

    = .    ANSWER

Write the given quadratic equation in the standard form ax^2 + bx + c = 0

    x^2 - 6x + 21 = 0.


The discriminant is

    d = b^2 - 4ac = (-6)^2 - 4*1*21 = 36 - 84 = -48.


Since the discriminant is negative, we conclude that the given equation 
has two different roots that are complex numbers.

I have been working on this math problem and I can't seem to figure it out. I was wondering if someone could help me? 
Please and Thank You!! I would deeply appreciate it!!

Find all intercepts of the parabola y-2=-1/2(x-4)^2

TOTALLY, TOTALLY UNNECESSARY steps by the person who responded.


 ----- Adding 2 to both sides
 ----- Vertex form of a parabola, where the VERTEX, (h, k) = (4, 2)
"h" is the x-coordinate of the vertex of the parabola, is also the paeabola's Axis of Symmetry, as well as the MIDWAY point
between the x-intercepts.
If MiDWAY between 2 points of a HORIZONTAL line is 4, then LEFT and RIGHT endpoints are: , and .
Therefore, x-intercepts are 2 and 6, or at the coordinate points: (2, 0) and (6, 0).


Although more time-consuming, we can also do the following:

 ---- Adding 2 to both sides

 ---- Substituting 0 for y, since y-values on x-axis are 0
- 2(0) =  ---- Multiplying each side by - 2

(x - 2)(x - 6) = 0 ---- Factorizing the trinomial
x - 2 = 0       OR      x - 6 = 0
x = 2           OR      x = 6
x-INTERCEPTS: (2, 0) (6, 0)

Your equation is

    12000*(1+r)^2 = 27000,


Divide both sides by 12000 to get

    (1+r)^2 = .


It gives

    (1+r)^2 = 2.25.


Take square root of both sides

    1 + r =  = 1.5.   <<<---===  it is not an approximate value  of the square root;
                                        it is the PRECISE value.


Hence,  r = 1.5 - 1 = 0.5.


Thus the problem is just solved completely.


        The ANSWER  is:  the annual rate of compounding is 0.5, or 50%.

If  (y-3)^2 = 16, then there are two cases.


(1)  Either y-3 =  = 4,   and then  y-3 = 4,  giving  y = 4+3 = 7,      y^2 = 49;


(2)  Or     y-3 =  = -4, and then  y-3 = -4, giving  y = -4 + 3 = -1,  y^2 = (-1)^2 = 1.


y^2 = 1 is less than y^2 = 49.  Therefore, the answer to the problem is THIS:


     +---------------------------------------------+
     |   The smallest possible value of y^2 is 1.  |
     +---------------------------------------------+

If a polynomial with real coefficients has a root which is a complex number, 
then this polynomial has another root which is the conjugate complex number.


It is a general property of polynomials with real coefficients.  
In our case, it means that the seeking polynomial has the roots  4i,  -4i  and  1.


Hence, the seeking polynomial is

    P(x) = (x-1)*(x-4i)*(x-(-4i)) = (x-1)*(x^2+16) = x^3 - x^2 + 16x - 16.


ANSWER.  P(x) = x^3 - x^2 + 16x - 16.

What would be the solution for x in the rational equation:

x^2/(x-3)= (x+2)/(2x-5)


 ---- Cross-multiplying

 <===== As the polynomial's LARGEST exponent is 3, there will be 3 solutions!!

Using the RATIONAL ROOT THEOREM, we find that 2 is a solution/zero of the above trinomial, so x = 2, or x - 2 = 0.
Hence, x - 2 is a factor of the polynomial.

Using SYNTHETIC DIVISION, or LONG DIVISION of POLYNOMIALS, the other factor of the polynomial is the quadratic, , so the factors of  are: .
As  CANNOT be factored with INTEGERS, its solutions/zeroes can be found, either by COMPLETING the SQUARE
or using the QUADRATIC EQUATION formula. Using the latter, we get: 

 <==== QUADRATIC EQUATION formula, where: 


                           

                            

The 3 solutions/zeroes (1 RATIONAL and 2 IRRATIONAL):   ;    ;  

On your TI-84 graphing calculator,

1. Press Y=
2. Make the screen read:

      Plot1 Plot2 Plot3 
     /Y1  =(2X+5)/(X+3)
     /Y2  =12/(X-1)

3. Press ZOOM then 6

See this graph:



Notice there are two points where the curves intersect. I have 
drawn circles around the two points.  We want the x-coordinate 
of the circled point on the right for it will have a larger 
x-value.

4. Press 2ND TRACE 5

Use the right arrow key to move the cursor to the right as close
as possible to the point.

5. Press ENTER ENTER ENTER.

Read X=7.3059371 at the bottom of the screen.  
That's the approximate answer.

Edwin