powerpointpresentation showing steps in factorizing a quadratic exprssion by completig the squares

1. M ODELING C OMPLETING THE S QUARE
Use algebra tiles to complete a perfect square trinomial.
2
Model the expression x +
6x. x2 x x x x x x
Arrange the x-tiles to
form part of a square.
x 1 1 1
To complete the square,
x 1 1 1
add nine 1-tiles. x 1 1 1
You have completed the square. x2 + 6x + 9 = (x + 3)2

2. S OLVING BY C OMPLETING THE S QUARE
To complete the square of the expression x2 + bx, add the
square of half the coefficient of x.
2 2
b = x+ b
x2 + bx + ( ) (
2 2 )

3. Completing the Square
What term should you add to x2 – 8x so that the result is a
perfect square?
S OLUTION
2
–8
The coefficient of x is –8, so you should add
to the expression.
( 2) , or 16,
2
–8
2
x – 8x +
2 ( ) = x2 – 8x + 16 = (x – 4)2

4. Completing the Square
Factor 2x2 – x – 2 = 0
S OLUTIO
N
2x2 – x – 2 = 0 Write original equation.
2x2 – x = 2 Add 2 to each side.
21
x – 2 x=1 Divide each side by 2.
2 1 1
2 1 2 1
2
x –
1
2
x+ –
1
( )
4
=1+
1
16
Add –
( •
2 2 ) ( 4)
= – , or
16
to each side.

5. Completing the Square
2 1 1
2 1 2 1
2
x –
1
2
x+ –
1
4 ( ) =1+
1
16
Add –
(•
2 2 ) ( 4)
= – , or
16
to each side.
2
1 17
(x –
4 ) =
16
Write left side as a fraction.
x–
1
= ±
17 Find the square root of each side.
4 4
1
± 17 Add
1 to each side.
x=
4 4 4
1 17 1 17
The solutions are + ≈ 1.28 and – ≈ – 0.78.
4 4 4 4

6. C HOOSING A S OLUTION M ETHOD
Investigating the Quadratic Formula
Perform the following steps on the general quadratic equation
ax2 + bx + c = 0 where a ≠ 0.
ax2 + bx = – c Subtract c from each side.
bx –c Divide each side by a.
x2 + a + = a
bx b 2 –c b 2
2
x +
a
+ ( )
2a ( )
= a +
2a
Add the square of half the coefficient
of x to each side.
b
x + ) = –a + b
c2 2
( 2a 4a 2 Write the left side as a perfect square.
b 2 – 4ac + b 2 Use a common denominator to express
( x+
2a ) = 2
4a the right side as a single fraction.

7. C HOOSING A S OLUTION M ETHOD
Investigating the Quadratic Formula
b 2 – 4ac + b 2 Use a common denominator to express
(x+
2a ) = 2
4a the right side as a single fraction.
± b − 4ac
2
b Find the square root of each side.
x+ =
2a 2a Include ± on the right side.
± b2 − 4ac b Solve for x by subtracting the same
x= –
2a 2a term from each side.
–b ± b − 4ac
2
Use a common denominator to express
x= the right side as a single fraction.
2a