HOW TO SOLVE QUADRATIC EQUATIONS - SOLVED EXAMPLES, EXERCISE

Please study
about Quadratic Equation before How To Solve Quadratic Equations
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 : How To Solve Quadratic Equations

Solve 4 + x - 14x² = 0.

Solution of Example 1 of How To Solve Quadratic Equations :

Factoring of the LHS of the Equation is shown below.

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Let P = 4 + x - 14x² = -14x² + x + 4

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

Step 1: Coefficient of x² x constant term
= -14 x 4 = -56

Step 2: We have to express -56 as two factors whose sum
= coefficient of x = 1;
-56 = 8 x (-7); [8 + (-7) = 1]

Step 3: P = -14x² + x + 4 = -14x² + 8x - 7x + 4

Step 4: P = 2x(-7x + 4) + 1(-7x + 4)

Step 5: P = (-7x + 4)(2x + 1)

Thus, By Factoring the Quadratic Polynomial, 4 + x - 14x²,we get the Factors as (-7x + 4)(2x + 1)

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4 + x - 14x² = 0 ⇒ (-7x + 4)(2x + 1) = 0
⇒ (-7x + 4) = 0 or (2x + 1) = 0
(-7x + 4) = 0 ⇒ -7x = -4 ⇒ x = -4⁄-7 = 4⁄7
(2x + 1) = 0 ⇒ 2x = -1 ⇒ x = -1⁄2
Thus, x = 4⁄7, -1⁄2 are the two roots of the given Quadratic equation. Ans.

Example 2 : How To Solve Quadratic Equations

Solve x² + 7x - 78 = 0.

Solution of Example 2 of How To Solve Quadratic Equations :

Factoring the LHS of the Equation is shown below.

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Let P = x² + 7x - 78

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

Step 1: Coefficient of x² x constant term
= 1 x -78 = -78

Step 2: We have to express -78 as two factors whose sum
= coefficient of x = 7 ;
-78 = -2 x 39 = -2 x 3 x 13 = -6 x 13; (-6 + 13 = 7)

Step 3: P = x² + 7x - 78 = x² - 6x + 13x - 78

Step 4: P = x(x - 6) + 13(x - 6)

Step 5: P = (x - 6)(x + 13)

Thus, By Factoring the Quadratic Polynomial, x² + 7x - 78,we get the Factors as (x - 6)(x + 13)

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x² + 7x - 78 = 0 ⇒ (x - 6)(x + 13) = 0
⇒ (x - 6) = 0 or (x + 13) = 0
(x - 6) = 0 ⇒ x = 6
(x + 13) = 0 ⇒ x = -13
Thus, x = 6, -13 are the two roots of the given Quadratic equation. Ans.

Exercise : How To Solve Quadratic Equations

1. Solve 2 + x - 15x² = 0.
2. Solve x² + 9x - 136 = 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 : How To Solve Quadratic Equations

1. 2⁄5, -1⁄3
2. -17, 8