Consider the general form of the quadratic expression given below.

ax2 + bx + c

The following steps will be helpful to fctor a quadratic expression in the above form.

Step 1 :

Multiply the coefficient of x2, a and the constant term c.

= ac

Step 2 :

Find two numbers p and q such that the product is equal to ac and the sum is equal to the coefficient of x, b.

pq = ac

p + q = b

Step 3 :

Split the middle term bx using the two numbers p and q.

ax2 + px + qx + c

Step 4 :

Factor the expression in step 3 by grouping as shown below.

= (ax2 + px) + (qx + c)

Factor each of the following quadratic expressions.

Example 1 :

x2 + 5x + 6

Solution :

Step 1 :

In the quadratic expression x2 + 5x + 6, the coefficient of x2 is 1 and the constant term is 6.

Multiply 1 and 6.

= 1 x 6

= 6

Step 2 :

Find two numbers such that the product is equal to 6 and the sum is equal to the coeffient of x, 5.

The two numbers satisfy the above condition are 2 and 3.

Step 3 :

Split the middle term 5x using the two numbers 2 and 3.

= x2 + 2x + 3x + 6

Step 4 :

Factor the expression in the above step 3 by grouping.

= (x2 + 2x) + (3x + 6)

= x(x + 2) + 3(x + 2)

= (x + 2)(x + 3)

Example 2 :

x2 + 17x + 60

Solution :

Step 1 :

In the quadratic expression x2 + 17x + 60, the coefficient of xis 1 and the constant term is 60.

Multiply 1 and 60.

= 1 x 60

= 60

Step 2 :

Find two numbers such that the product is equal to 60 and the sum is equal to the coeffient of x, 17.

The two numbers satisfy the above condition are 5 and 12.

Step 3 :

Split the middle term 5x using the two numbers 5 and 12.

= x2 + 5x + 12x + 60

Step 4 :

Factor the expression in the above step 3 by grouping.

= (x2 + 5x) + (12x + 60)

= x(x + 5) + 12(x + 5)

= (x + 5)(x + 12)

Example 3 :

x2 - 5x - 24

Solution :

Step 1 :

In the quadratic expression x2 - 5x - 24, the coefficient of xis 1 and the constant term is 24.

Multiply 1 and -24.

= 1 x (-24)

= -24

Step 2 :

Find two numbers such that the product is equal to -24 and the sum is equal to the coeffient of x, -5.

The two numbers satisfy the above condition are -8 and 3.

Step 3 :

Split the middle term -5x using the two numbers -8 and 3.

= x2 - 8x + 3x - 24

Step 4 :

Factor the expression in the above step 3 by grouping.

= (x2 - 8x) + (3x - 24)

= x(x - 8) + 3(x - 8)

= (x - 8)(x + 3)

Example 4 :

x2 - 2x + 1

Solution :

Step 1 :

In the quadratic expression x2 - 2x + 1, the coefficient of xis 1 and the constant term is 1.

Multiply 1 and 1.

= 1 x 1

= 1

Step 2 :

Find two numbers such that the product is equal to 1 and the sum is equal to the coeffient of x, -2.

The two numbers satisfy the above condition are -1 and -1.

Step 3 :

Split the middle term -2x using the two numbers -1 and -1.

= x2 - x - x + 1

Step 4 :

Factor the expression in the above step 3 by grouping.

= (x2 - x) + (-x + 1)

= x(x - 1) - 1(x - 1)

= (x - 1)(x - 1)

Example 5 :

2x2 + 11x + 12

Solution :

Step 1 :

In the quadratic expression 2x2 + 11x + 12, the coefficient of xis 2 and the constant term is 12.

Multiply 2 and 12.

= 2 x 12

= 24

Step 2 :

Find two numbers such that the product is equal to 24 and the sum is equal to the coeffient of x, 11.

The two numbers satisfy the above condition are 3 and 8.

Step 3 :

Split the middle term 11x using the two numbers 3 and 8.

= 2x2 + 3x + 8x + 12

Step 4 :

Factor the expression in the above step 3 by grouping.

= (2x2 + 3x) + (8x + 12)

= x(2x + 3) + 4(2x + 3)

= (2x + 3)(x + 4)

Example 6 :

2x2 + x - 6

Solution :

Step 1 :

In the quadratic expression 2x2 + x - 6, the coefficient of x2 is 2 and the constant term is -6.

Multiply 2 and -6.

= 2 x (-6)

= -12

Step 2 :

Find two numbers such that the product is equal to -12 and the sum is equal to the coeffient of x, 1.

The two numbers satisfy the above condition are -3 and 4.

Step 3 :

Split the middle term x using the two numbers 3 and 8.

= 2x2 - 3x + 4x - 6

Step 4 :

Factor the expression in the above step 3 by grouping.

= (2x2 - 3x) + (4x - 6)

= x(2x - 3) + 2(2x - 3)

= (2x - 3)(x + 2)

Example 7 :

3x2 – 5x – 12

Solution :

Step 1 :

In the quadratic expression 3x2 – 5x – 12, the coefficient of xis 3 and the constant term is -12.

Multiply 3 and -12.

= 3 x (-12)

= -36

Step 2 :

Find two numbers such that the product is equal to -36 and the sum is equal to the coeffient of x, -5.

The two numbers satisfy the above condition are -9 and 4.

Step 3 :

Split the middle term -5x using the two numbers -9 and 4.

= 3x2 – 9x + 4x – 12

Step 4 :

Factor the expression in the above step 3 by grouping.

= (3x2 – 9x) + (4x – 12)

= 3x(x - 3) + 4(x - 3)

= (x - 3)(3x + 4)

Example 8 :

x2 - 25

Solution :

The quadratic expression x2 – 25 can be factored using the following algebraic identity.

a2 - b2 = (a + b)(a - b)

= x2 – 25

= x2 – 52

= (x + 5)(x - 5)

Example 9 :

16x2 - 9

Solution :

= 16x2 - 9

= 42x2 – 32

= (4x)– 32

= (4x + 3)(4x - 3)

Example 10 :

(x + 3)2 - 81

Solution :

= (x + 3)2 - 81

= (x + 3)2 – 92

= (x + 3 + 9)(x + 3 - 9)

= (x + 12)(x - 6)

Kindly mail your feedback to v4formath@gmail.com

## Recent Articles

1. ### Simplifying Algebraic Expressions with Fractional Coefficients

May 17, 24 08:12 AM

Simplifying Algebraic Expressions with Fractional Coefficients

2. ### The Mean Value Theorem Worksheet

May 14, 24 08:53 AM

The Mean Value Theorem Worksheet