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AC Method: Factoring Quadratic Trinomials Using the AC Method

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

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 the 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?

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 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, the result would be 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:

Steps in Using the AC Method 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, 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:

2x2 grid for the AC test

2x2 grid for the AC test

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.

 2x2 grid for the AC test

2x2 grid for the AC test

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

 2x2 grid for the AC test

2x2 grid for the AC test

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. Use the trial and error method to 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 2x2 grid and fill it out with the correct terms.

AC method for quadratic trinomials where C is positive

AC method for quadratic trinomials where C is positive

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 method for quadratic trinomials where C Is positive

AC method for quadratic trinomials where C Is positive

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. Use the trial and error method to 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 2x2 grid and fill it out with the correct terms.

AC method for quadratic trinomials where C is negative

AC method for quadratic trinomials where C is negative

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 method for quadratic trinomials where C is negative

AC method for quadratic trinomials where C is negative

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. Use the trial and error method to 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 2x2 grid and fill it out with the correct terms.

AC method for quadratic trinomials where C is positive

AC method for quadratic trinomials where C is positive

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 method for quadratic trinomials where C is positive

AC method for quadratic trinomials where C is positive

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).

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