10.5 completing the square. 10.5 – completing the square goals / “i can…” solve quadratic...
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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 33
different ways:GraphingSquare RootsFactoring
y = x2 – 4x – 5
Solutions are
-1 and 5
10.5 – Completing the Square
How many solutions are there? What are they?
1. 25x2 = 16 ANSWER 4 5
4 5– ,
2. 9m2 = 100 ANSWER 103– ,
103
3. 49b2 + 64 = 0 ANSWER no solution
10.5 – Completing the Square
Use the Square Root method to solve:
Example 1
x2 – 2x – 24 = 0
(x + 4)(x – 6) = 0
x + 4 = 0 x – 6 = 0
x = –4 x = 6
Example 2
x2 – 8x + 11 = 0
x2 – 8x + 11 is prime; therefore, another method must be used to solve this equation.
10.5 – Completing the Square
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
x + 8x + 16 is factored into
(x + 4) notice that the 4 is (½ * 8)
2
2 2
10.5 – Completing the Square
This is ALWAYSALWAYS the case with perfect squares. The last term in the binomial can be found by the formula ½ b
Using this idea, we can make polynomials that aren’t perfect squares into perfect squares.
2
10.5 – Completing the Square
Example:
x + 22x + ____ What number
would fit in the
last term to make
it a perfect
square?
2
10.5 – Completing the Square
(½ * 22) = 121
SO….. x + 22x + 121 should be a
perfect square.
(x + 11)
2
2
2
10.5 – Completing the Square
What numbers should be added to each equation to complete the square?
x + 20x
x - 8x
x + 50x
2
2
2
This method will work to solve ALL quadratic equations;
HOWEVER
it is “messymessy” to solve quadratic equations by completing the square if a ≠ 1a ≠ 1 and/or b is an odd number.
Completing the square is a GREATGREAT 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 evena = 1, b is even
x2 – 6x - 7 = 0
x2 – 6x + 9 = 7 + 9
(x – 3)2 = 16
x – 3 = ± 4
x = 7 OR 1
Example 2
a ≠ 1, b is not even
3x2 – 5x + 2 = 0
2 5 2 03 3
x x
2 5 25 2 253 36 3 36
x x
25 16 36
x
5 16 6
x
5 16 6
x
5 16 6
x
OR
x = 1 OR x = ⅔
10.5 – Completing the Square
10.5 – Completing the Square
Solving x + bx = c
x + 8x = 48 I want to solve
this using perfect
squares.
How can I make the left side of the equation a perfect square?
2
2
10.5 – Completing the Square
Use ½ b (½ * 8) = 16Add 16 to both sides of the equation. (we
MUSTMUST keep the equation equivalent)
x + 8x + 16 = 48 + 16Make the left side a perfect square
binomial.
(x + 4) = 64
2 2
2
10.5 – Completing the Square
x + 4 = 8SO……….
x + 4 = 8 x + 4 = -8
x = 4 x = -12
+-
10.5 – Completing the Square
Solving x + bx + c = 0
x + 12x + 11 = 0 Since it is not aperfect square,move the 11 tothe other side.
x + 12x = -11 Now, can youcomplete the
squareon the left side?
2
2
2
Find the value of cc 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
3. x2 + 3x + c
ANSWER 36; (x 6)2
ANSWER ; (x )294
32
10.5 – Completing the Square
Solve x2 – 16x = –15 by completing the square.
SOLUTION
Write original equation.x2 – 16x = –15
Add , or (– 8)2, to each side.
– 16 2
2x2 – 16x + (– 8)2 = –15 + (– 8)2
Write left side as the square of a binomial.
(x – 8)2 = –15 + (– 8)2
Simplify the right side.(x – 8)2 = 49
10.5 – Completing the Square
Take square roots of each side.x – 8 = ±7
Add 8 to each side.x = 8 ± 7
ANSWER
The solutions of the equation are 8 + 7 = 15 and 8 – 7 = 1.
10.5 – Completing the Square
10.5 – Completing the Square
x + 12x + ? = -11 + ?
x + 12x + = -11 +
(x + ) =
2
2
2
10.5 – Completing the Square
Complete the square
x - 20x + 32 = 02
10.5 – Completing the Square
Complete the square
x + 3x – 5 = 02
10.5 – Completing the Square
Complete the square
x + 9x = 1362
10.5 – Completing the Square
Still a little foggy?If so, watch this video to see if it will
help