Completing the square is a method that can be used to solve quadratic equations. It involves rearranging the quadratic into a form which can make it easier to solve. It is another way of solving a quadratic equation, instead of factorising or using the quadratic formula.
Complete the square
To complete the square on a quadratic expression, it should be put into the form
(x+a)2+b
Once the quadratic has been put in the completed square form, it can be rearranged to solve for x.
PROCEDURE
1.
Write the quadratic expression in descending powers of x.
2.
Set up the answer in the form (x)2.
3.
Take the co-efficient of x from the quadratic, half the number and fill this number in the bracket.
4.
Take the number that has just been filled in the bracket, square the number and subtract this number from the bracket.
5.
Add the constant term.
Example 1
Complete the square on the quadratic expression
x2+6x
Start by setting up the answer in the form
(x)2
Take the co-efficient ofx, in this case6 and half the number, which gives3. Fill this number in the bracket.
(x+3)2
Take the number in the bracket,3, and square it. Subtract this number from the bracket. This gives
(x+3)2−9
Example 2
Complete the square on the quadratic expression
x2+10x+6
Start by setting up the answer in the form
(x)2
Take the co-efficient ofx,in this case10and half the number, which gives5.Fill this number in the bracket.
(x+5)2
Take the number in the bracket,5,and square it. Subtract this number from the bracket and add the constant term. This gives
(x+5)2−25+6(x+5)2−19
Solve a quadratic by completing the square
Once a quadratic has been rearranged into the completed square form, it is possible to solve the equation.
Example 3
Solve
x2+10x+6=0
This has already been rearranged into the correct form in example 2 above.
(x+5)2−19=0
To solve the quadratic, take19to the other side of the equation and then square root, before taking5over.
(x+5)2−19(x+5)2x+5x=0=19=±19=−5±19
The answer can be left in surd form, or can be calculated to3.s.f.