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Solving Quadratic Equations by
Completing the Square
Perfect Square Trinomials
Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36
These quadratic equations are PERFECT SQUARE TRINOMIALS (three terms) because the factors are perfect squares. Try factoring these out then check.
(x+3)(x+3) (x-5)(x-5) (x+6)(x+6)
Creating a Perfect Square Trinomial
In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____
Find the constant term by squaring half the coefficient of the linear term.
(14/2)2
X2 + 14x + 49
Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___
100
4
25/4
However, not all trinomials are perfect squares trinomials.
Example: 2x2+13x+15 = (2x+3)(x+5)(2x+3)(x+5) are not perfect squares.
But there is a way to “fix” ALL trinomials to make them a Perfect Square Trinomial
Solving Quadratic Equations by Completing the Square
Solve the following equation by completing the square:
Step 1: Move quadratic term, and linear term to left side of the equation
2 8 20 0x x
2 8 20x x
Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.
2 8 =20 + x x 21
( ) 4 then square it, 4 162
8
2 8 2016 16x x What have we done here? If a is 1, we just took the half of b and squared it. Then we added that number to the left and the right side.
Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
2 8 2016 16x x
2
( 4)( 4) 36
( 4) 36
x x
x
We can then use the previous completed square to solve the quadratic equation.
Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
2( 4) 36x
( 4) 6x
Solving Quadratic Equations by Completing the Square
Step 5: Set up the two possibilities and solve
4 6
4 6 an
d 4 6
10 and 2 x=
x
x x
x
Completing the Square-Example #2
Solve the following equation by completing the square:
Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.
22 7 12 0x x
22 7 12x x
Observe that for this problem a=2. Our approach will have to be modified.
Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.
First, the quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.
2
2
2
2 7
2
2 2 2
7 12
7
2
=-12 +
6
x x
x x
xx
21 7 7 49
( ) then square it, 2 62 4 4 1
7
2 49 49
16 1
76
2 6x x
Next, take half of b, square it then add to both sides.
Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
2
2
2
76
2
7 96 49
4 16 16
7 47
4
49 49
16 1
16
6x x
x
x
Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
27 47( )
4 16x
7 47( )
4 4
7 47
4 4
7 47
4
x
ix
ix
Why is there an i? i represents a complex number. This allows us to take the square root of a negative number. If you’re taking the square root of a negative number, just take the square root then add an i.
Solving Quadratic Equations by Completing the Square
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1. 9,7
2.(6, 14)
3. 3,8
7 34.
2
5 475.
6
i
i
