# AC Method: Factoring Quadratic Trinomials Using the AC Method

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## 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 the form

*AX*where

^{2}+ BX + C*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*and the factors are

^{2}- 2xy + y^{2}*(x + y)*and

^{2}*(x - y)*, respectively. On the other hand, the general quadratic trinomial is a form

^{2}*Ax*where

^{2}+ Bx + C*A*may stand for any integer. But how do you easily factor quadratic trinomials?

## What Is the AC Method?

The 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, the result would be

*B*. For example, let us apply the AC test in factoring

*3x*. In the given trinomial, the product of

^{2}+ 11x + 10*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 Method 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, the result would be

*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*.

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

c. Create a 2x2 grid and fill it out with the correct 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*are

^{2}- 17x + 5*(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*.

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

c. Create a 2x2 grid and fill it out with the correct 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*are

^{2}- 17x - 14*(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.

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

c. Create a 2x2 grid and fill it out with the correct 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*are

^{2}- 17x - 14*(2x + 1)*and

*(2x + 3)*.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2018 Ray**