SOLVING QUADRATIC EQUATIONS BY FACTORING - FACTORING, ROOTS, EXAMPLES, EXERCISE

Please study
about Quadratic Equation before Solving Quadratic Equations by Factoring,
if you have not already done so.

There, we gave introduction to Quadratic polynomial,
Quadratic Equation, Methods to solve the Quadratic
Equations and about Method of Solving by Factoring.

That knowledge is a prerequisite here.

Here, we apply the Method to solve problems.
Solved Examples and Exercise problems are given.

Example 1 : Solving Quadratic Equations by Factoring

Solve 9x² + 26x + 16 = 0

Solution of Example 1 of Solving Quadratic Equations by Factoring :

The Factoring of the LHS of the above Equation is given below.

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Let P = 9x² + 26x + 16

In Factoring of Trinomials (Quadratics) , we follow the five steps.

Step 1: Coefficient of x² x constant term
= 9 x 16 = 144

Step 2: We have to express 144 as two factors whose sum
= coefficient of x = 26;
144 = 2 x 72 = 2 x 2 x 36 = 2 x 2 x 2 x 18 = 8 x 18; (8 + 18 = 26)

Step 3: P = 9x² + 26x + 16 = 9x² + 8x + 18x+ 16

Step 4: P = x(9x + 8) + 2(9x + 8)

Step 5: P = (9x + 8)(x + 2)

Thus, Factoring the Quadratic Polynomial, 9x² + 26x + 16,
we get the Factors as (9x + 8)(x + 2).

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9x² + 26x + 16 = 0 ⇒ (9x + 8)(x + 2) = 0
⇒ (9x + 8) = 0 or (x + 2) = 0
(9x + 8) = 0 ⇒ 9x = -8 ⇒ x = -8⁄9
(x + 2) = 0 ⇒ x = -2
Thus, x = -8⁄9, -2 are the two roots of the given Quadratic equation. Ans.

Example 2 : Solving Quadratic Equations by Factoring

Solve 2 - 5x - 18x² = 0

Solution of Example 2 of Solving Quadratic Equations by Factoring :

The Factoring of the LHS of the above Equation is given below.

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Let P = 2 - 5x - 18x² = -18x² - 5x + 2

In Factoring of Trinomials (Quadratics) , we follow the five steps.

Step 1: Coefficient of x² x constant term
= -18 x 2 = -36

Step 2: We have to express -36 as two factors whose sum
= coefficient of x = -5 ;
-36 = -2 x 18 = -2 x 2 x 9 = 4 x -9; [4 + (-9) = -5]

Step 3: P = -18x² - 5x + 2 = -18x² + 4x - 9x+ 2

Step 4: P = 2x(-9x + 2) + 1(-9x + 2)

Step 5: P = (-9x + 2)(2x + 1)

Thus, Factoring of Quadratic Polynomial,2 - 5x - 18x²,
we get the Factors as (-9x + 2)(2x + 1).

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2 - 5x - 18x² = 0 ⇒ (-9x + 2)(2x + 1) = 0
⇒ (-9x + 2) = 0 or (2x + 1) = 0
(-9x + 2) = 0 ⇒ -9x = -2 ⇒ x = -2⁄-9 = 2⁄9
(2x + 1) = 0 ⇒ 2x = -1 ⇒ x = -1⁄2
Thus, x = 2⁄9, -1⁄2 are the two roots of the given Quadratic equation. Ans.

Exercise : Solving Quadratic Equations by Factoring

1. Solve 8x² + 25x + 18 = 0.
2. Solve 3 - 2x - 21x² = 0.

For Answers See at the bottom of the Page.

NOTE: You may solve these problems of Exercise
using Quadratic Formula, after learning it.

Answers to Exercise : Solving Quadratic Equations by Factoring

1. -9⁄8, -2
2. -3⁄7, 1⁄3