FACTORING QUADRATIC EXPRESSIONS

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 x² is 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 x² is 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 x² is 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 x² is 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 x² is 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)² – 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

We always appreciate your feedback.