# Factoring Quadratic Trinomials Using the AC Test

## What Is a Trinomial?

The expression x^{2} - 5x + 7 is a trinomial. It is a trinomial expression because it contains three terms. Trinomial expressions are in a form AX^{2} + BX + C where A, B, and C are integers. The four major types of trinomial expressions are:

1. Trinomial squares

2. Quadratic trinomials of the form AX^{2} + BX + C, where C is positive

3. Quadratic trinomials of the form AX^{2} + BX + C, where C is negative

4. General quadratic trinomials with coefficients

Trinomial Squares are trinomials in which the first term and the third term are both squares and positive. The form of a trinomial square is either x^{2} + 2xy + y^{2} or x^{2} - 2xy + y^{2} and the factors are (x + y)^{2} and (x - y)^{2}, respectively. On the other hand, the general quadratic trinomial is a form Ax^{2} + Bx + C where A may stand for any integer. But how do you easily factor quadratic trinomials?

## What Is AC Test?

AC test is a method of testing whether a quadratic trinomial is factorable or not. It is also a method of identifying the factors of a general quadratic trinomial Ax^{2} + B(x) + C. A quadratic trinomial is factorable if the product of A and C have M and N as two factors such that when added would result to B. For example, let us apply the AC test in factoring 3x^{2} + 11x + 10. In the given trinomial, the product of A and C is 30. Then, find the two factors of 30 that will produce a sum of 11. The answer would be 5 and 6. Hence, the given trinomial is factorable. Once the trinomial is factorable, solve for the factors of the trinomial. Here are the steps in using the AC test in factoring trinomials.

## Steps in Using the AC Test in Factoring Quadratic Trinomials

1. From the quadratic trinomial Ax^{2} + B(x) + C, multiply A and C. Then, find the two factors of A and C such that when added would result to B.

M = first factor

N = first factor

M + N = B

2. If the trinomial is factorable, proceed to the AC test. Prepare a two by two grid and label each from 1 to 4. Construct like the one below.

3. Given an expression Ax^{2} + B(x) + C, place the first term of the trinomial in 1 and the third term in 3. Place M and N in grids 2 and 4, respectively. To check, the products of diagonal terms must be the same.

4. Factor each row and column. Once factored, combine the answers.

## Problem 1: Quadratic Trinomials Where C Is Positive

Apply the AC test in factoring 6x^{2} - 17x + 5.

**Solution**

a. Solve for AC. Multiply the coefficient A by the coefficient C.

A = 6 C = 5 AC = 6 X 5 AC = 30

b. By trial and error method, solve for the factors of 30 that will give -17.

M = -15 N = -2 M + N = -17 -15 - 2 = -17 -17 = -17

c. Create a two by two grid and fill it out with the right terms.

d. Factor each row and column.

Columns:

a. The common factor of 6(x)^{2} and -2(x) is 2(x).

b. The common factor of -15(x) and 5 is -5.

Rows:

a. The common factor of 6(x)^{2} and -15(x) is 3(x).

b. The common factor of -2(x) and 5 is -1.

**Final Answer:** The factors of trinomials in a form x^{2} + bx + c are (x + r) and (x - s). The factors of the equation 6x^{2} - 17x + 5 are (2x - 5) and (3x - 1).

## Problem 2: Quadratic Trinomials Where C Is Negative

Apply the AC test in factoring 6x^{2} - 17x - 14.

**Solution**

a. Solve for AC. Multiply the coefficient A by the coefficient C.

A = 6 C = -14 AC = 6 X -14 AC = -84

b. By trial and error method, solve for the factors of -84 that will give -17.

M = -21 N = 4 M + N = -17 -21 + 4 = -17 -17 = -17

c. Create a two by two grid and fill it out with the right terms.

d. Factor each row and column.

Columns:

a. The common factor of 6(x)^{2} and 4(x) is 2(x).

b. The common factor of -21(x) and -14 is -7.

Rows:

a. The common factor of 6(x)^{2} and -21(x) is 3(x).

b. The common factor of 4(x) and -14 is 2.

**Final Answer: **The factors of trinomials in a form x^{2} + bx + c are (x + r) and (x - s). The factors of 6x^{2} - 17x - 14 are (3x + 2) and (2x - 7).

## Problem 3: Quadratic Trinomials Where C Is Positive

Apply the AC test in factoring 4x^{2} + 8x + 3.

**Solution**

a. Solve for AC. Multiply the coefficient A by the coefficient C.

A = 4 C = 3 AC = 4 X 3 AC = 12

b. By trial and error method, solve for the factors of 12 that will give 8.

M = 6 N = 2 M + N = 8 2 + 6 = 8 8 = 8

c. Create a two by two grid and fill it out with the right terms.

d. Factor each row and column.

Columns:

a. The common factor of 4(x)^{2} and 2(x) is 2(x).

b. The common factor of 6(x) and 3 is 3.

Rows:

a. The common factor of 4(x)^{2} and 6(x) is 2(x).

b. The common factor of 2(x) and 3 is 1.

**Final Answer: **The factors of trinomials in a form x^{2} + bx + c are (x + r) and (x + s).** ** The factors of 6x^{2} - 17x - 14 are (2x + 1) and (2x + 3).

## Quiz About AC Test

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## Questions & Answers

**© 2018 John Ray**

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