Solving Quadratic Equations by Completing the Square
Those quadratic equations that are not written in the
completed square form can be rewritten using the following
technique.
Completing the Square
The technique of completing the square uses algebra to put any
quadratic equation in completed square form. To understand how to
write a quadratic expression of the form x^{ 2} + bx as a
perfect square, work backwards.
( x + k )^{ 2} = x^{ 2} + 2 kx + k^{ 2}
Notice that the coefficient of x in this expression is 2k .
So, to write x^{ 2} + bx as a perfect square trinomial x^{
2} + bx + c , let b = 2 k , or .
Example 1
Write x^{ 2} + 12x as a perfect square trinomial x^{
2} + 12x + c , where c is a constant.
Solution
First, find . In this quadratic
expression, b = 12. or 6.
Next, square this result. 6^{ 2} = 36
Finally, add the result to the original expression. x ^{2}
+ 12x + 36
This is a perfect square trinomial since x ^{2} + 12x
+ 36 = ( x + 6) ^{2} .
Completing the square is very helpful in solving quadratic
equations.
Example2
Solve x ^{2} + 6 x + 3 = 0.
Solution
Since x ^{2} + 6 x + 3 is not a perfect square,
subtract 3 from each side and then complete the square of the
quadratic expression.
x ^{2} + 6 x + 3 = 0
x ^{2} + 6 x = 3 Subtract 3 from each side.
First, find . In the expression x ^{2}
+ 6 x , b = 6. or 3
Next, square this result. 3 ^{2 }= 9
Finally, add the result to each side of the equation. x ^{2}
+ 6 x + 9 = 3 + 9
Now simplify.
x ^{2} + 6 x + 9 = 3 + 9
( x + 3) ^{2} = 6 Factor x ^{2} + 6x + 9.
This equation is in completed square form. Now solve by taking
the square root of each side.
The method of completing the square will also let you know
when there are no solutions to a quadratic equation, as shown in
the following example.
Example 3
Solve x ^{2}  x + 2 = 0.
Solution
Since the expression is not a perfect square, begin by
subtracting 2 from each side of the equation to get x ^{2}
 x = 2. Then complete the square of the expression x ^{2}
 x .
First, find . In this expression, b
= 1.
Next, square this result.
Finally, add the result to each side of the equation.
Now simplify.
The expression is always greater than
or equal to zero because the square of any number is greater than
or equal to zero. However, the righthand side of this equation, is negative. Therefore,
there can be no solutions to this equation. Thus, the original
equation x ^{2}  x + 2 = 0 has no real solution.
Try to find out what it means graphically when there are no
real solutions to a quadratic equation ax ^{2} + bx + c =
0. It means that the graph of the quadratic function y = ax ^{2}
+ bx + c (a parabola) does not intersect the xaxis.
Proving the Quadratic Formula
We have seen that the method of completing the square is very
useful in solving quadratic equations. It is also important in
proving the Quadratic Formula.
