Solve a Quadratic Equation by Factoring

SOLVE A QUADRATIC EQUATION BY FACTORING

Solve a Quadratic Equation by Factoring :

In this section, you will learn how to solve quadratic equations by factoring.

Generally we have two types of quadratic equation.

Quadratic equation of leading coefficient 1.
Quadratic equation of leading coefficient not equal to 1.

The general form of a quadratic equation is

ax² + bx + c = 0

In a quadratic equation, leading coefficient is nothing but the coefficient of x².

Solving Quadratic Equations by Factoring with a Leading Coefficient of 1 - Procedure

(i) In a quadratic equation in the form ax² + bx + c = 0, if the leading coefficient is 1, we have to decompose the constant term "c" into two factors.

(ii) The product of the two factors must be equal to the constant term "c" and the addition of two factors must be equal to the coefficient of x, that is "b".

(iii) If p and q are the two factors of the constant term c, then we have to factor the quadratic equation using p and q as shown below.

(x + p)(x + q) = 0

(iv) Solving the above equation, we get

x = -p and x = -q

How to assign signs for the two factors ?

Quadratic Equation

Signs of Factors

ax² + bx + c = 0

Positive sign for both the factors.

ax² - bx + c = 0

Negative sign for both the factors.

ax² + bx - c = 0

Negative sign for smaller factor and positive sign for larger factor.

ax² - bx - c = 0

Positive sign for smaller factor and negative sign for larger factor.

Solve a quadratic equation by factoring - Examples

Example 1 :

Solve the following quadratic equation by factoring :

x² + 9x + 14 = 0

Solution :

In the given quadratic equation, the coefficient of x² is 1.

Decompose the constant term +14 into two factors such that the product of the two factors is equal to +14 and the addition of two factors is equal to the coefficient of x, that is +9.

Then, the two factors of +14 are

+7 and +2

Factor the given quadratic equation using +7 and +2 and solve for x.

(x + 7)(x + 2) = 0

x + 7 = 0 or x + 2 = 0

x = -7 or x = -2

So, the solution is {-7, -2}.

Example 2 :

Solve the following quadratic equation by factoring :

x² - 9x + 14 = 0

Solution :

In the given quadratic equation, the coefficient of x² is 1.

Decompose the constant term +14 into two factors such that the product of the two factors is equal to +14 and the addition of two factors is equal to the coefficient of x, that is -9.

Then, the two factors of +14 are

-7 and -2

Factor the given quadratic equation using -7 and -2 and solve for x.

(x - 7)(x - 2) = 0

x - 7 = 0 or x - 2 = 0

x = 7 or x = 2

So, the solution is {7, 2}.

Example 3 :

Solve the following quadratic equation by factoring :

x² + 5x - 14 = 0

Solution :

In the given quadratic equation, the coefficient of x² is 1.

Decompose the constant term -14 into two factors such that the product of the two factors is equal to -14 and the addition of two factors is equal to the coefficient of x, that is +5.

Then, the two factors of -14 are

+7 and -2

Factor the given quadratic equation using +7 and -2 and solve for x.

(x + 7)(x - 2) = 0

x + 7 = 0 or x - 2 = 0

x = -7 or x = 2

So, the solution is {-7, 2}.

Example 4 :

Solve the following quadratic equation by factoring :

x² - 5x - 14 = 0

Solution :

In the given quadratic equation, the coefficient of x² is 1.

Decompose the constant term -14 into two factors such that the product of the two factors is equal to -14 and the addition of two factors is equal to the coefficient of x, that is -5.

Then, the two factors of -14 are

+2 and -7

Factor the given quadratic equation using +2 and -7 and solve for x.

(x + 2)(x - 7) = 0

x + 2 = 0 or x - 7 = 0

x = -2 or x = 7

So, the solution is {-2, 7}.

Example 5 :

Solve the quadratic equation by factoring :

3x² – 5x – 12 = 0

Solution :

In the given quadratic equation, the coefficient of x² is not 1.

So, multiply the coefficient of x² and the constant term "-12".

3 ⋅ (-12) = -36

Decompose -36 into two factors such that the product of two factors is equal to -36 and the addition of two factors is equal to the coefficient of x, that is -5.

Then, the two factors of -36 are

+4 and -9

Now we have to divide the two factors 4 and -3 by the coefficient of x², that is 3.