# 2.1.1: Methods for Solving Quadratic Functions

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## Factoring Polynomials in Quadratic Form

The volume of a rectangular prism is 10x3−25x2−15x. What are the lengths of the prism's sides?

## Factoring Polynomials in Quadratic Form

The last type of factorable polynomial are those that are in quadratic form. Quadratic form is when a polynomial looks like a trinomial or binomial and can be factored like a quadratic. One example is when a polynomial is in the form ax4+bx2+c. Another possibility is something similar to the difference of squares, a4−b4. This can be factored to (a2−b2)(a2+b2) or (a−b)(a+b)(a2+b2). Always keep in mind that the greatest common factors should be factored out first.

Let's factor the following polynomials.

1. 2x4−x2−15

This particular polynomial is factorable. First, ac=−30. The factors of -30 that add up to -1 are -6 and 5. Expand the middle term and then use factoring by grouping.

2x4−x2−15

2x4−6x2+5x2−15

2x2(x2−3)+5(x2−3)

(x2−3)(2x2+5)

Both of the factors are not factorable, so we are done.

2. 81x4−16

Treat this polynomial equation like a difference of squares.

81x4−16

(9x2−4)(9x2+4)

Now, we can factor 9x2−4 using the difference of squares a second time.

(3x−2)(3x+2)(9x2+4)

9x2+4 cannot be factored because it is a sum of squares. This will have imaginary solutions.

Now, let's find all the real-number solutions of 6x5−51x3−27x=0.

First, pull out the GCF among the three terms.

6x5−51x3−27x=0

3x(2x4−17x2−9)=0

Factor what is inside the parenthesis like a quadratic equation. ac=−18 and the factors of -18 that add up to -17 are -18 and 1. Expand the middle term and then use factoring by grouping.

6x5−51x3−27x=0

3x(2x4−17x2−9)=0

3x(2x4−18x2+x2−9)=0

3x[2x2(x2−9)+1(x2−9)]=0

3x(x2−9)(2x2+1)=0

Factor x2−9 further and solve for x where possible. 2x2+1 is not factorable.

3x(x2−9)(2x2+1)=0

3x(x−3)(x+3)(2x2+1)=0

x=−3,0,3

## Examples

###### Example 1

Earlier, you were asked to find the lengths of the prism's sides.

Solution

To find the lengths of the prism's sides, we need to factor 10x3−25x2−15x.

First, pull out the GCF among the three terms.

10x3−25x2−15x

5x(2x2−5x−3)

Factor what is inside the parenthesis like a quadratic equation. ac=−6 and the factors of -6 that add up to -5 are -6 and 1.

5x(2x2−5x−3)=5x(2x+1)(x−3)

Therefore, the lengths of the rectangular prism's sides are 5x, 2x+1, and x−3.

###### Example 2

Factor: 3x4+14x2+8.

Solution

ac=24 and the factors of 24 that add up to 14 are 12 and 2.

3x4+14x2+8

3x4+12x2+2x2+8

3x2(x2+4)+2(x4+4)

(x2+4)(3x2+2)

###### Example 3

Factor: 36x4−25.

Solution

Factor this polynomial like a difference of squares.

36x4−25

(6x2−5)(6x2+5)

6 and 5 are not square numbers, so this cannot be factored further.

###### Example 4

Find all the real-number solutions of 8x5+26x3−24x=0.

Solution

Pull out a 2x from each term.

8x5+26x3−24x=0

2x(4x4+13x−12)=0

2x(4x4+16x2−3x2−12)=0

2x[4x2(x2+4)−3(x2+4)]=0

2x(x2+4)(4x2−3)=0

Set each factor equal to zero.

4x2−3=0

2x=0

x2+4=0

and x2= $$\ 3 \over 4$$

x=0

x2=−4

x=± $$\ \frac{\sqrt{3}}{2}$$

Notice the second factor will give imaginary solutions.

## Review

1. x4−6x2+8
2. x4−4x2−45
3. x4−18x2+45
4. 4x4−11x2−3
5. 6x4+19x2+8
6. x4−81
7. 16x4−1
8. 6x5+26x3−20x
9. 4x6−36x2
10. 625−81x4

Find all the real-number solutions to the polynomials below.

1. 2x4−5x2−12=0
2. x4−16=0
3. 16x4−49=0
4. 12x6+69x4+45x2=0
5. 3x4+17x2−6=0

## Vocabulary

Term Definition
Factor to Solve "Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of x that make each binomial equal to zero.
factored form The factored form of a quadratic function f(x) is f(x)=a(x−r1)(x−r2), where r1 and r2 are the roots of the function.
Factoring Factoring is the process of dividing a number or expression into a product of smaller numbers or expressions.
Quadratic form A polynomial in quadratic form looks like a trinomial or binomial and can be factored like a quadratic expression.
quadratic function A quadratic function is a function that can be written in the form f(x)=ax2+bx+c, where a, b, and c are real constants and a≠0.
Roots The roots of a function are the values of x that make y equal to zero.
standard form The standard form of a quadratic function is f(x)=ax2+bx+c.
Vertex form The vertex form of a quadratic function is y=a(x−h)2+k, where (h,k) is the vertex of the parabola.
Zeroes of a Polynomial The zeroes of a polynomial f(x) are the values of x that cause f(x) to be equal to zero.

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