Download Completing The Square

MODELING COMPLETING THE SQUARE
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
x x x x x x
x
x
x
1
1
1
1
1
1
1
1
1
x2 + 6x + 9 = (x + 3)2
SOLVING BY COMPLETING THE SQUARE
To complete the square of the expression x2 + bx, add the
square of half the coefficient of x.
x2 + bx +
2
( ) (
b
2
=
x+ b
2
2
)
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
to the expression.
–8
2
x – 8x +
2
2
–8
, or 16,
( 2)
2
( )=x
2
– 8x + 16 = (x – 4)2
Completing the Square
Factor 2x2 – x – 2 = 0
SOLUTION
2x2 – x – 2 = 0
Write original equation.
2x2 – x = 2
Add 2 to each side.
x2 – 1 x = 1
Divide each side by 2.
2
x2 –
2
( )
1
1
x+ –
2
4
=1+
1
16
(
1 1
Add –
•
2 2
to each side.
2
1 2
1
= –
, or
) ( 4)
16
Completing the Square
x2 –
2
( )
1
1
x+ –
2
4
(
1
x–
4
x–
2
)
=1+
(
1 1
Add –
•
2 2
1
16
x=
16
Write left side as a fraction.
17
4
1

4
1
The solutions are
+
4
) ( 4)
to each side.
17
=
16
1
= 
4
2
1 2
1
= –
, or
Find the square root of each side.
17
4
Add
1 to each side.
4
1
17
–
 1.28 and
4
4
17
 – 0.78.
4
Completing the Square
1
The solutions are
+
4
CHECK
1
17
–
 1.28 and
4
4
17
 – 0.78.
4
You can check the solutions on a graphing calculator.
CHOOSING A SOLUTION METHOD
Investigating the Quadratic Formula
Perform the following steps on the general quadratic equation
ax2 + bx + c = 0 where a  0.
ax2 + bx = – c
x2
Subtract c from each side.
bx
–c
+ a+ = a
2
Divide each side by a.
2
( )
( )
b
–c + b
x
+
=
( 2a) a 4a
bx
b
x2 + +
a
2a
–c
= a +
2
b
2a
2
2
(
b
x+
2a
2
)
Add the square of half the coefficient
of x to each side.
–
4ac + b 2
=
2
4a
Write the left side as a perfect square.
Use a common denominator to express
the right side as a single fraction.
CHOOSING A SOLUTION METHOD
Investigating the Quadratic Formula
(
b
x+
2a
2
)
2
–
4ac
+
b
=
2
4a

b
x+
=
2a
2
b - 4ac
2a
 b2 - 4ac b
x=
–
2a
2a
x=
2
–b  b - 4ac
2a
Use a common denominator to express
the right side as a single fraction.
Find the square root of each side.
Include ± on the right side.
Solve for x by subtracting the same
term from each side.
Use a common denominator to express
the right side as a single fraction.