Factoring Quadratic Trinomials Using the AC Test

Updated on July 9, 2018
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JR has a Bachelor of Science in Civil Engineering and specializes in Structural Engineering. He loves to write anything about education.

What Is a Trinomial?

The expression x2 - 5x + 7 is a trinomial. It is a trinomial expression because it contains three terms. Trinomial expressions are in a form AX2 + 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 AX2 + BX + C, where C is positive

3. Quadratic trinomials of the form AX2 + 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 x2 + 2xy + y2 or x2 - 2xy + y2 and the factors are (x + y)2 and (x - y)2, respectively. On the other hand, the general quadratic trinomial is a form Ax2 + Bx + C where A may stand for any integer. But how do you easily factor quadratic trinomials?

Factoring Quadratic Trinomials Using AC Test
Factoring Quadratic Trinomials Using AC Test | Source

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 Ax2 + 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 3x2 + 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.

Factoring Quadratic Trinomials Using AC Test
Factoring Quadratic Trinomials Using AC Test | Source

Steps in Using the AC Test in Factoring Quadratic Trinomials

1. From the quadratic trinomial Ax2 + 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.

2 x 2 Grid for AC Test
2 x 2 Grid for AC Test | Source

3. Given an expression Ax2 + 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.

2 x 2 Grid for AC Test
2 x 2 Grid for AC Test | Source

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

2 x 2 Grid in AC Test
2 x 2 Grid in AC Test | Source

Problem 1: Quadratic Trinomials Where C Is Positive

Apply the AC test in factoring 6x2 - 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.

AC Test for Quadratic Trinomials Where C Is Positive
AC Test for Quadratic Trinomials Where C Is Positive | Source

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.

AC Test for Quadratic Trinomials Where C Is Positive
AC Test for Quadratic Trinomials Where C Is Positive | Source

Final Answer: The factors of trinomials in a form x2 + bx + c are (x + r) and (x - s). The factors of the equation 6x2 - 17x + 5 are (2x - 5) and (3x - 1).

Problem 2: Quadratic Trinomials Where C Is Negative

Apply the AC test in factoring 6x2 - 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.

AC Test for Quadratic Trinomials Where C is Negative
AC Test for Quadratic Trinomials Where C is Negative | Source

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.

AC Test for Quadratic Trinomials Where C is Negative
AC Test for Quadratic Trinomials Where C is Negative | Source

Final Answer: The factors of trinomials in a form x2 + bx + c are (x + r) and (x - s). The factors of 6x2 - 17x - 14 are (3x + 2) and (2x - 7).

Problem 3: Quadratic Trinomials Where C Is Positive

Apply the AC test in factoring 4x2 + 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.

AC Test for Quadratic Trinomials Where C Is Positive
AC Test for Quadratic Trinomials Where C Is Positive | Source

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.

AC Test for Quadratic Trinomials Where C Is Positive
AC Test for Quadratic Trinomials Where C Is Positive | Source

Final Answer: The factors of trinomials in a form x2 + bx + c are (x + r) and (x + s). The factors of 6x2 - 17x - 14 are (2x + 1) and (2x + 3).

Quiz About AC Test

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    © 2018 John Ray

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