Solving Quadratic Equations
22.3 Solution by Completing the Square
We have seen that expressions of the form :
a^{2}x^{2} − b^{2}
are known as differences of squares and can be factorised as follows:
(ax − b)(ax + b).
This simple factorisation leads to another technique to solve quadratic
equations known as
completing the square.
We demonstrate with a simple example, by trying to solve for x in :
x^{2} − 2x − 1 = 0.(22.1)
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} − 2x + 1.
However, we can cheat and create a perfect square by adding 2 to both sides of
the equation in
(22.1) as:
Now we know that:
which means that:
(x − 1)^{2} − 2
is a difference of squares. Therefore we can write:
he solution to x^{2} − 2x − 1 = 0 is then:
or
This means or
. This example demonstrates the use of
completing the
square to solve a quadratic equation .
Method : Solving Quadratic Equations by Completing the
Square
1. Write the equation in the form ax^{2} + bx + c = 0. e.g. x^{2}
+ 2x . 3 = 0
2. Take the constant over to the right hand side of the equation. e.g. x^{2}
+ 2x = 3
3. If necessary, make the coefficient of the x ^{2} term = 1, by dividing
through by the existing
coefficient.
4. Take half the coefficient of the x term, square it and add it to both sides
of the equation.
e.g. in x^{2} + 2x = 3, half of the x term is 1. 1^{1} = 1.
Therefore we add 1 to both sides to
get: x^{2} + 2x + 1 = 3 + 1.
5. Write the left hand side as a perfect square: (x + 1)^{2}  4 = 0
6. You should then be able to factorise the equation in terms of difference of
squares and
then solve for x: (x + 1  2)(x + 1 + 2) = 0
Worked Example 106: Solving
Quadratic Equations by Completing the Square Question: Solve: x^{2} − 10x − 11 = 0 by completing the square Answer Step 1 : Write the equation in the form ax^{2} + bx + c = 0 x^{2} − 10x − 11 = 0 Step 2 : Take the constant over to the right hand side of the equation x^{2} − 10x = 11 Step 3 : Check that the coefficient of the x^{2} term is 1. The coefficient of the x^{2} term is 1. Step 4 : Take half the coefficient of the x term, square it and add it to both sides The coefficient of the x term is . Therefore: x^{2} − 10x + 25 = 11 + 25 Step 5 : Write the left hand side as a perfect square (x − 5)^{2} − 36 = 0 Step 6 : Factorise equation as difference of squares (x − 5)^{2} − 36 = 0 [(x − 5) + 6][(x − 5) − 6] = 0 Step 7 : Solve for the unknown value [x + 1][x − 11] = 0 x = −1 or x = 11 

Worked Example 107: Solving
Quadratic Equations by Completing the Square Question: Solve: 2x^{2} − 8x − 16 = 0 by completing the square Answer Step 1 : Write the equation in the form ax^{2} + bx + c = 0 2x^{2} − 8x − 16 = 0 Step 2 : Take the constant over to the right hand side of the equation 2x^{2} − 8x = 16 Step 3 : Check that the coefficient of the x^{2} term is 1. The coefficient of the x^{2} term is 2. Therefore, divide both sides by 2: x^{2} − 4x = 8 Step 4 : Take half the coefficient of the x term, square it and add it to both sides The coefficient of the x term is . Therefore: x^{2} − 4x + 4 = 8 + 4 Step 5 : Write the left hand side as a perfect square (x − 2)^{2} − 12 = 0 Step 6 : Factorise equation as difference of squares Step 7 : Solve for the unknown value Step 8 : The last three steps can also be done in a different the way Leave left hand side written as a perfect square (x − 2)^{2}= 12 Step 9 : Take the square root on both sides of the equation Step 10 : Solve for x Therefore or Compare to answer in step 7. 

Exercise: Solution by Completing the Square Solve the following equations by completing the square: 
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