Completing the Square & Solving Quadratic Equations by Completing
the Square
After this lesson you should be able to determine the roots of a quadratic equation by the method of completing the
square.
A perfect square is anything times itself…look at the following perfect square trinomials and what they become in
factored form.
Perfect square trinomial
Factored as a perfect square
X
2
+ 8x + 16
(x+4)(x+4) = (x + 4)
2
X
2
(x + 6)(x + 6) = (x + 6)
2
X
2
–
18x + 81
( x
–
9)(x
–
9) = (x
–
9)
2
X
2
–
40x + 400
(x
–
20)(x
–
20) = (x
–
20)
2
What pattern do you notice in all the perfect square trinomials (hint: the pattern is with the __ and __ terms when
written in standard form)?
Is the following trinomial a perfect square? X
2
– 6x – 16 _______ because ________________________________.
Is the following trinomial a perfect square? X
2
+ 16x + 64 _______ because ________________________________.
What would the c-term need to be in the following trinomial to be a perfect square? X
2
– 20x + C
Now factor that trinomial.
Steps for
Steps for Steps for
Steps for SOLVING by
SOLVING by SOLVING by
SOLVING by Completing the Square:
Completing the Square:Completing the Square:
Completing the Square:
1. Be sure that the coefficient of the highest power is one.
If it is not, divide each term by that value to create a
leading coefficient of one.
2. Move the constant term to the right hand side.
3. Prepare to add the needed value to create the perfect
square trinomial. Be sure to balance the equation. The
boxes may help you remember to balance.
4. To find the needed value for the perfect square
trinomial, take half of the coefficient of the middle term (x-
term), square it, and add that value to both sides of the
equation.
5. Factor the perfect square trinomial.
6. Take the square root of each side and solve. Remember
to consider both plus and minus results.
When written in simplest radical form we would get…
X = -
√
20
– 4 AND X = +
√
20
– 4
X = -2
√
5
−
4

+
2
√
5
−
4
Follow the Exact steps from the notes above to solve by completing the square.
X
2
+ 6x + 1 = 0
X
2
–
10x
–
4
= 0
X
2
+ 12x + 8
= 0