PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Solving quadratics by factoring
Quadratics
Passport to advanced math

LEARNING OBJECTIVE

After this lesson, you will be able to:

· Solve a quadratic equation by factoring

To answer a question like this:

If x² + x = 20 and x < 0, what is the value of x − 7?

A. −12

B. −5

C. 5

D. 12

You need to know this:

A quadratic expression is a second-degree polynomial—that is, a polynomial containing a squared variable. You can write a quadratic expression as ax² + bx + c.

The FOIL acronym (which stands for First, Outer, Inner, Last) will help you remember how to multiply two binomials: multiply the first terms together (ac), then the outer terms (ad), then the inner terms (bc), and finally the last terms (bd):

(a + b)(c + d) = ac + ad + bc + bd

FOIL can also be done in reverse if you need to go from a quadratic to its factors.

To solve a quadratic equation by factoring, the quadratic must be set equal to zero. For example:

From the binomial factors, you can find the solutions, also called roots or zeros, of the equation. For two factors to be multiplied together and produce zero as the result, one or both those factors must be zero. In the example above, either x + 8 = 0 or x − 7 = 0, which means that x = −8 or x = 7.

You need to do this:

To solve a quadratic equation by factoring:

· Set the quadratic equal to zero, so it looks like this: ax² + bx + c = 0.

· Factor the squared term. (For factoring, it’s easiest when a, the coefficient in front of x², is equal to 1.)

· Make a list of the factors of c. Remember to include negatives.

· Find the factor pair that, when added, equals b, the coefficient in front of x.

· Write the quadratic as the product of two binomials.

· Set each binomial equal to zero and solve.

Explanation:

Set the equation equal to zero and factor the first term:

Next, consider factors of −20, keeping in mind that they must sum to 1, so the factor with the greater absolute value must be positive. The possibilities are 20 × −1, 10 × −2 and 5 × −4. The factor pair that sums to 1 is 5 × −4. Write that factor pair into your binomials:

Set each factor equal to zero and solve:

The question says that x < 0, so x = −5. However, you are not done. The question asks for x − 7, which is −12. Therefore, (A) is correct.

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

1. Which of the following is an equivalent form of the expression (x − 4)(x + 2)?