Solving Quadratic Equations by the Formulae

The method of completing the square can be best used to establish a formula for the solution of the quadratic equation of the form 

\( \scriptsize ax^2 \: +\: bx\: +\: c = 0 \)

The formula is given as:

x = \(\large \frac{-b \pm \sqrt {b^2 \: – \: 4ac}}{2a}\)

The formula above is known as the quadratic formula, where

a is the coefficient of \( \scriptsize x^2 \)

b = coefficient of x

c = constant term.

Note that, the quadratic formula is particularly used to solve quadratic equations, which cannot be solved using factorization method.

Example 5.2.1:

Solve the following equations and leave your answers in 2decimal place (d.p) using the quadratic formula.

i. \( \scriptsize 3x^2 \: – \: 5x \: – \: 9 = 0 \)

ii. \( \scriptsize – 3x^2 = (x\:+\:2)(x\:-\:1) \)

iii. \( \scriptsize \left(\scriptsize 2x \:- \: 3 \right)^2 = x^2 \)

iv. \( \scriptsize -4x^2 \: + \: 2x \: + \: 6 = 0 \)

Solution:

i. \( \scriptsize 3x^2 \: – \: 5x \: – \: 9 = 0 \)

Let a = 3, b = -5 and c = -9

Using the factor formula we have: