SOLVING QUADRATIC EQUATIONS BY FACTORING

To solve a quadratic equation by factoring, the quadratic equation has ro be in standard form, that is

ax2 + bx + c = 0

In the above standard form of a quadratic equation,

coefficient of x2 = a

coefficient of x = b

constant term = c

The following steps can be used to solve a quadratic equation by factoring.

Step 1 :

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

= ac

Step 2 :

Resolve ac into two factors, say m and n such that the product of them is equal to ac and the sum is equal to the coefficient of x, b.

mn = ac

m + n = b

Step 3 :

Split the middle term bx using the two numbers m and n.

ax2 + mx + nx + c = 0

Step 4 :

Factor and solve the quadratic equation in step 3 by grouping as shown below.

(ax2 + mx) + (nx + c) = 0

Solve each of the following quadratic equations by factoring.

Example 1 :

x2 + 5x + 6 = 0

Solution :

Step 1 :

In the quadratic equation x2 + 5x + 6 = 0, multiply the coefficient of x2, 1 and the constant term 6.

= 1 x 6

= 6

Step 2 :

Resolve 6 into two factors such that the product of them is equal to 6 and the sum is equal to the coeffient of x, 5.

2 x 3 = 6

2 + 3 = 5

So, the two factors of 6 are 2 and 3.

Step 3 :

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

x2 + 2x + 3x + 6 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(x2 + 2x) + (3x + 6) = 0

x(x + 2) + 3(x + 2) = 0

(x + 2)(x + 3) = 0

x + 2 = 0  or  x + 3 = 0

x = -2  or  x = -3

Example 2 :

x2 + 10x + 24 = 0

Solution :

Step 1 :

In the quadratic equation x2 + 10x + 24 = 0, multiply the coefficient of x2, 1 and the constant term 24.

= 1 x 24

= 24

Step 2 :

Resolve 24 into two factors such that the product of them is equal to 24 and the sum is equal to the coeffient of x, 10.

4 x 6 = 24

4 + 6 = 10

So, the two factors of 24 are 4 and 6.

Step 3 :

Split the middle term 10x using the two factors 4 and 6.

x2 + 4x + 6x + 24 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(x2 + 4x) + (6x + 24) = 0

x(x + 4) + 6(x + 4) = 0

(x + 4)(x + 6) = 0

x + 4 = 0  or  x + 6 = 0

x = -4  or  x = -6

Example 3 :

x2 - 7x + 10 = 0

Solution :

Step 1 :

In the quadratic equation x2 - 7x + 10 = 0, multiply the coefficient of x2, 1 and the constant term 10.

= 1 x 10

= 10

Step 2 :

Resolve 10 into two factors such that the product of them is equal to 10 and the sum is equal to the coeffient of x, -7.

(-5) x (-2) = 10

(-5) + (-2) = -7

So, the two factors of 10 are -5 and -2.

Step 3 :

Split the middle term -7x using the two factors -5 and -2.

x2 - 5x - 2x + 10 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(x2 - 5x) + (-2x + 10) = 0

x(x - 5) - 2(x - 5) = 0

(x - 5)(x - 2) = 0

x - 5 = 0  or  x - 2 = 0

x = 5  or  x = 2

Example 4 :

x2 + 4x - 12 = 0

Solution :

Step 1 :

In the quadratic equation x2 + 4x - 12 = 0, multiply the coefficient of x2, 1 and the constant term -12.

= 1 x (-12)

= -12

Step 2 :

Resolve -12 into two factors such that the product of them is equal to -12 and the sum is equal to the coeffient of x, 4.

-2 x 6 = -12

-2 + 6 = 4

So, the two factors of -12 are -2 and 6.

Step 3 :

Split the middle term 4x using the two factors -2 and 6.

x2 - 2x + 6x - 12 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(x2 - 2x) + (6x - 12) = 0

x(x - 2) + 6(x - 2) = 0

(x - 2)(x + 6) = 0

x - 2 = 0  or  x + 6 = 0

x = 2  or  x = -6

Example 5 :

x2 - x - 6 = 0

Solution :

Step 1 :

In the quadratic equation x2 - x - 6 = 0, multiply the coefficient of x2, 1 and the constant term -6.

= 1 x (-6)

= -6

Step 2 :

Resolve -6 into two factors such that the product of them is equal to -6 and the sum is equal to the coeffient of x, -1.

-3 x 2 = -6

-3 + 2 = -1

So, the two factors of -6 are -3 and 2.

Step 3 :

Split the middle term -x using the two factors -3 and 2.

x2 - 3x + 2x - 6 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(x2 - 3x) + (2x - 6) = 0

x(x - 3) + 2(x - 3) = 0

(x - 3)(x + 2) = 0

x - 3 = 0  or  x + 2 = 0

x = 3  or  x = -2

Example 6 :

2x2 + 7x + 6 = 0

Solution :

Step 1 :

In the quadratic equation 2x2 + 7x + 6 = 0, multiply the coefficient of x2, 2 and the constant term 6.

= 2 x 6

= 12

Step 2 :

Resolve 12 into two factors such that the product of them is equal to 12 and the sum is equal to the coeffient of x, 8.

3 x 4 = 12

3 + 4 = 7

So, the two factors of 12 are 3 and 4.

Step 3 :

Split the middle term 7x using the two factors 3 and 4.

2x2 + 3x + 4x + 6 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(2x2 + 3x) + (4x + 6) = 0

x(2x + 3) + 2(2x + 3) = 0

(2x + 3)(x + 2) = 0

2x + 3 = 0  or  x + 2 = 0

x = ⁻³⁄₂  or  x = -2

Example 7 :

-2x2 - 2x + 5 = -5x2 + 13

Solution :

Step 1 :

The given quadratic equation is not in standard form. Write it in standard form.

-2x2 - 2x + 5 = -5x2 + 13

Add 5xto both sides.

3x2 - 2x + 5 = 13

Subtract 13 from both sides.

3x2 - 2x - 8 = 0

Step 2 :

In the quadratic expression 3x2 - 2x - 8 = 0, multiply the coefficient of x2, 3 and the constant term -8.

= 3 x (-8)

= -24

Step 3 :

Resolve -24 into two factors such that the product of them is equal to -24 and the sum is equal to the coeffient of x, -2.

-6 x 4 = -24

-6 + 4 = -2

So, the two factors of 12 are -6 and 4.

Step 3 :

Split the middle term -2x using the two factors -6 and 4.

3x2 - 6x + 4x - 8 = 0

Step 4 :

Factor and solve the quadratic equation in the above step 3 by grouping.

(3x2 - 6x) + (4x - 8) = 0

3x(x - 2) + 4(x - 2) = 0

(x - 2)(3x + 4) = 0

x - 2 = 0  or  3x + 4 = 0

x = 2  or  x = ⁻⁴⁄₃

Example 8 :

x2 - 25 = 0

Solution :

The quadratic equation x2 - 25 = 0 can be solved using the following algebraic identity.

a2 - b2 = (a + b)(a - b)

x2 - 25 = 0

x2 - 52 = 0

(x + 5)(x - 5) = 0

x + 5 = 0  or  x - 5 = 0

x = -5  or  x = 5

Example 9 :

4x2 - 49 = 0

Solution :

4x2 - 49 = 0

42x2 – 7= 0

(4x)– 7= 0

(4x + 7)(4x - 7) = 0

4x + 7 = 0  or  4x - 7 = 0

x = -⁷⁄₄  or  x = ⁷⁄₄

Example 10 :

(x - 2)2 - 36 = 0

Solution :

(x - 2)2 - 36 = 0

(x - 2)2 – 6= 0

(x - 2 + 6)(x - 2 - 6) = 0

(x + 4)(x - 8) = 0

x + 4 = 0  or  x - 8 = 0

x = -4  or  x = 8

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