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Factoring, completing the square, and the quadratic formula.

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Practice Methods for Solving Quadratic Functions
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The volume of a rectangular prism is . What are the lengths of the prism's sides?

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 . Another possibility is something similar to the difference of squares, . This can be factored to or . Always keep in mind that the greatest common factors should be factored out first.

Factor

Factor .

This particular polynomial is factorable. Let’s use the method we learned in the Factoring When the Leading Coefficient Doesn't Equal 1 concept. First, . The factors of -30 that add up to -1 are -6 and 5. Expand the middle term and then use factoring by grouping.

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

Factor .

Treat this polynomial equation like a difference of squares.

Now, we can factor using the difference of squares a second time.

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

Find all the real-number solutions of .

First, pull out the GCF among the three terms.

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

Factor further and solve for where possible. is not factorable.

Examples

Example 1

Earlier, you were asked what are the lengths of the prism's sides.

To find the lengths of the prism's sides, we need to factor .

First, pull out the GCF among the three terms.

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

Therefore, the lengths of the rectangular prism's sides are , , and .

Factor the following polynomials.

Example 2

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

Example 3

Factor this polynomial like a difference of squares.

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

Example 4

Find all the real-number solutions of .

Pull out a from each term.

Set each factor equal to zero.

Notice the second factor will give imaginary solutions.

Review

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

To see the Review answers, open this PDF file and look for section 6.8.

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Vocabulary Language: English

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 $f(x)=a(x-r_{1})(x-r_{2})$, where $r_{1}$ and $r_{2}$ 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.

A polynomial in quadratic form looks like a trinomial or binomial and can be factored like a quadratic expression.

A quadratic function is a function that can be written in the form $f(x)=ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.

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 $f(x)=ax^{2}+bx+c$.

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.