Quadratic Equations

When solving a quadratic equation, we need to bear in mind that if the product of two numbers is equal to zero, then at least one of the numbers or both numbers must be zero.

e.g. if xy = 0, then either x = 0 or y = 0 or both x = 0 and y = 0.

To solve the quadratic equation of the form ax² + bx + c = 0 when \( \scriptsize a \neq 0, b \neq 0, c \neq 0 \! \! \! \!\)  

First, find the product of the coefficient of x² and the constant term c.

i.e. a × c = ac.

Then find two factors of ac that add up to b (i.e. the coefficient of x).

Factorise the LHS of the equation to obtain two factors. Put each factor equal to 0 and then solve the two equations obtained.

Example 6.5.1: 

Use factorisation method to solve the following quadratic equations:

a. \( \scriptsize 2P = 35 \: – \: P^2 \)
b. \( \frac{1}{5}\scriptsize \:+ \:3a = \: -10a^2 \)
c. \( \scriptsize 29y \: – \: 14 = 12y^2 \)
d. \( \scriptsize 8x^3 \: – \: 44x^2\: + \:60x = 0 \)

Solution

a. \( \scriptsize 2P = 35 \: – \: P^2 \)

Rearrange as \( \scriptsize P^2 + 2P \: – \: 35 = 0 \)