m odeling c ompleting the s quare
DESCRIPTION
x. x. x. x. x. x. x 2. x. x. x. M ODELING C OMPLETING THE S QUARE. Use algebra tiles to complete a perfect square trinomial. Model the expression x 2 + 6 x. Arrange the x -tiles to form part of a square. 1. 1. 1. 1. 1. 1. To complete the square, add nine 1-tiles. 1. 1. - PowerPoint PPT PresentationTRANSCRIPT
x xx
MODELING COMPLETING THE SQUARE
11 1
11 1
11 1
x2 x x x
xxx
Use algebra tiles to complete a perfect square trinomial.
Model the expression x 2 + 6x.
Arrange the x-tiles to form part of a square.
To complete the square, add nine 1-tiles.
You have completed the square. x2 + 6x + 9 = (x + 3)2
SOLVING BY COMPLETING THE SQUARE
x2 + bx + = x + )( b2 ( )b
2
2 2
To complete the square of the expression x2 + bx, add the square of half the coefficient of x.
Completing the Square
What term should you add to x2 – 8x so that the result is a perfect square?
SOLUTION
The coefficient of x is –8, so you should add , or 16, to the expression.
)(–82
2
x2 – 8x + )(–82
2= x2 – 8x + 16 = (x – 4)2
Completing the Square
SOLUTION
Divide each side by 2.
2x2 – x = 2
Factor 2x2 – x – 2 = 0
2x2 – x – 2 = 0
x2 – x = 112
Write original equation.
Add 2 to each side.
12
x2 – x + = 1 +)( 1
4
2– 1
16Add = , or
( 1
2– •
12)
116
2
)( 1
4
2–
to each side.
Completing the Square
)1
4
2–(x =
1716
174
1
4x – =
12
x2 – x + = 1 +)( 1
4
2– 1
16Add = , or
( 1
2– •
12)
116
2
)( 1
4
2–
to each side.
Write left side as a fraction.
Find the square root of each side.
Add to each side. 14
x = 1
4 17
4
1
4The solutions are + 1.28 and – 0.78.
1
4–17
4174
Completing the Square
You can check the solutions on a graphing calculator.
1
4The solutions are + 1.28 and – 0.78.
1
4–17
4174
CHECK
CHOOSING A SOLUTION METHOD
Investigating the Quadratic Formula
Perform the following steps on the general quadratic equationax2 + bx + c = 0 where a 0.
ax2 + bx = – c
x2 + + = bxa
– ca
bxa
x2 + + = + b
2a)( – ca
2 b2a)(
2
b2a)( 2
x + = +– ca
b2
4a2
Use a common denominator to express the right side as a single fraction.
b2a)( 2
x + = – 4ac + b 2
4a2
Subtract c from each side.
Divide each side by a.
Add the square of half the coefficient of x to each side.
Write the left side as a perfect square.
CHOOSING A SOLUTION METHOD
b
2a
b 2 4ac
2ax + =
Use a common denominator to express the right side as a single fraction.
Investigating the Quadratic Formula
Use a common denominator to express the right side as a single fraction.
b2a)( 2
x + = – 4ac + b 2
4a 2
x = –b2a2a
b
2 4ac
x =2a
–b b
2 4ac
Find the square root of each side.Include ± on the right side.
Solve for x by subtracting the same term from each side.