CHAPTER 7. SOLVINGQUADRATIC EQUATIONS 7.3
7.3 Solution by Completing the Square
EMBAH
We have seen that expressions of the form:a^2 x^2 − b^2are known as differences of squares and can befactorised as follows:(ax− b)(ax + b).This simple factorisationleads to another technique to solve quadratic equations known as completing
the square.
We demonstrate with asimple example, by trying to solve for x in:x^2 − 2 x− 1 = 0. (7.1)We cannot easily find factors of this term, but the first two terms look similar to the first two terms of
the perfect square:
(x− 1)^2 = x^2 − 2 x + 1.
However, we can cheatand create a perfect square by adding 2 to both sides of the equation in (7.1)
as:x^2 − 2 x− 1 = 0
x^2 − 2 x− 1 + 2 = 0 + 2
x^2 − 2 x + 1 = 2
(x− 1)^2 = 2
(x− 1)^2 − 2 = 0Now we know that:
2 = (√
2)^2
which means that:
(x− 1)^2 − 2
is a difference of squares. Therefore we can write:(x− 1)^2 − 2 = [(x− 1)−√
2][(x− 1) +√
2] = 0.
The solution to x^2 − 2 x− 1 = 0 is then:(x− 1)−√
2 = 0
or
(x− 1) +√
2 = 0.
This means x = 1 +√
2 or x = 1−√
2. This example demonstrates the use of completing the square
to solve a quadratic equation.Method: Solving Quadratic Equations by Completing the Square- Write the equation inthe form ax^2 + bx + c = 0. e.g. x^2 + 2x− 3 = 0
- Take the constant over to the right hand side of the equation, e.g. x^2 + 2x = 3
- Make the coefficientof the x^2 term = 1, by dividing through bythe existing coefficient.
- Take half the coefficient of the x term, square it and addit to both sides of the equation, e.g. in
x^2 + 2x = 3, half of the coefficientof the x term is 1 and 12 = 1. Therefore we add 1 to both
sides to get: x^2 + 2x + 1 = 3 + 1.