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10.5
Completing the Square
10.5 – Completing the Square
Goals / “I can…”
Solve quadratic equations by completing
the square
10.5 – Completing the Square
Review:
Remember we’ve solved quadratics using 3
different ways:
Graphing
Square Roots
Factoring
10.5 – Completing the Square
How many solutions are there? What are they?
2
y = x – 4x – 5
Solutions are
-1 and 5
10.5 – Completing the Square
Use the Square Root method to solve:
1.
25x2 = 16
ANSWER
– 4
5
4
, 5
2.
9m2 = 100
ANSWER
– 10 , 10
3
3
3.
49b2 + 64 = 0
ANSWER
no solution
10.5 – Completing the Square
Example 1
2
x – 2x – 24 = 0
(x + 4)(x – 6) = 0
x+4=0 x–6=0
x = –4
x=6
Example 2
2
x – 8x + 11 = 0
2
x – 8x + 11 is
prime; therefore,
another method
must be used to
solve this equation.
10.5 – Completing the Square
The easiest trinomials to look at are
often perfect squares because they
always have the SAME characteristics.
10.5 – Completing the Square
2
x + 8x + 16
2
(x + 4)
is factored into
2
notice that the 4 is (½ * 8)
10.5 – Completing the Square
This is ALWAYS the case with perfect
squares. The last term in the binomial
2
can be found by the formula ½ b
Using this idea, we can make
polynomials that aren’t perfect squares
into perfect squares.
10.5 – Completing the Square
Example:
2
x + 22x + ____
What number
would fit in the
last term to make
it a perfect
square?
10.5 – Completing the Square
2
(½ * 22) = 121
2
SO….. x + 22x + 121 should be a
perfect square.
2
(x + 11)
10.5 – Completing the Square
What numbers should be added to each
equation to complete the square?
2
x + 20x
x 2 - 8x
2
x + 50x
10.5 – Completing the Square
 This method will work to solve ALL quadratic
equations;
HOWEVER
 it is “messy” to solve quadratic equations by
completing the square if a ≠ 1 and/or b is an odd
number.
 Completing the square is a GREAT choice for
solving quadratic equations if a = 1 and b is an
even number.
10.5 – Completing the Square
Example 1
a = 1, b is even
2
x – 6x - 7 = 0
2
x – 6x + 9 = 7 + 9
2
(x – 3) = 16
x–3=±4
x = 7 OR 1
Example 2
a ≠ 1, b is not even
3x2 – 5x + 2 = 0
x2  5 x  2  0
3 3
x2  5 x  25   2  25
3 36
3 36


5
x 

6 

2
1
36
x 5   1
6
6
x 51
6 6
OR
x 51
6 6
x = 1 OR x = ⅔
10.5 – Completing the Square
2
Solving x + bx = c
2
x + 8x = 48
I want to solve
this using perfect
squares.
How can I make the left side of the
equation a perfect square?
10.5 – Completing the Square
Use ½ b
2
2
(½ * 8) = 16
 Add 16 to both sides of the equation. (we
MUST keep the equation equivalent)
x + 8x + 16 = 48 + 16
Make the left side a perfect square
binomial.
2
(x + 4) = 64
10.5 – Completing the Square
x + 4 = +- 8
SO……….
x+4=8
x=4
x + 4 = -8
x = -12
10.5 – Completing the Square
Solving x 2 + bx + c = 0
x 2 + 12x + 11 = 0
x 2 + 12x = -11
Since it is not a
perfect square,
move the 11 to
the other side.
Now, can you
complete the square
on the left side?
10.5 – Completing the Square
Find the value of c that makes the expression a perfect
square trinomial. Then write the expression as the
square of a binomial.
1.
x2 + 8x + c
ANSWER
16; (x + 4)2
2.
x2  12x + c
ANSWER
36; (x  6)2
3.
x2 + 3x + c
ANSWER
9 ; (x 3 )2
4
2
10.5 – Completing the Square
Solve x2 – 16x = –15 by completing the square.
SOLUTION
x2 – 16x = –15
Write original equation.
– 16
2
2
2
x – 16x + (– 8) = –15 + (– 8) Add
2
2
, or (– 8)2, to
each side.
(x – 8)2 = –15 + (– 8)2 Write left side as the
square of a binomial.
(x – 8)2 = 49
Simplify the right side.
10.5 – Completing the Square
x – 8 = ±7
x=8±7
Take square roots of each side.
Add 8 to each side.
ANSWER
The solutions of the equation are 8 + 7 = 15 and 8 – 7 = 1.
10.5 – Completing the Square
2
x + 12x + ? = -11 + ?
2
x + 12x +
(x +
2
) =
= -11 +
10.5 – Completing the Square
Complete the square
2
x - 20x + 32 = 0
10.5 – Completing the Square
Complete the square
2
x + 3x – 5 = 0
10.5 – Completing the Square
Complete the square
2
x + 9x = 136
10.5 – Completing the Square
Still a little foggy?
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