Pc Solve By Quads By Comp Sq

Solve Qua dratics by Comple ting th e Squ are MATH PRO JECT

MODELING COMPLETING THE SQUARE Use algebra tiles to complete a perfect square trinomial. Model the expression x 2 + 6x. Arrange the x­tiles to  form part of a square. To complete the square,  add nine 1­tiles.

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 square of half the coefficient of x.

x2 + bx +

2

( ) ( b 2

x2 + bx, add the 

= x+ b 2

2

)

Completing the Square

What term should you add to x2 perfect square?

– 8x so that the result is a 

SOLUTION

The coefficient of x is –8, so you should add to the expression.

–8 x – 8x + 2 2

2 –8 , or 16,

( 2)

2

( )

= x2 – 8x + 16 = (x – 4)2

Completing the Square

Factor  2x2

–x–2=0

SOLUTION Write original equation.

2x2 – x – 2 = 0

Add 2 to each side.

2x2 – x = 2 1 x – 2 x=1

Divide each side by 2.

2

x2 –

2

( )

1 1 x+ – 2 4

=1+ 1

16

2 1 2 1 1 1 – – • Add                    =               , or 2 2               16 4

(

to each side.

)( )

Completing the Square

x2 –

( )

1 1 x+ – 2 4

(

1 x– 4

x–

2 2 1 1 1 – – • 1 Add                    =               , or 2 2               16 4

2

2

)

(

=1+ 1

16

to each side.

17 = 16

1 = ± 4

x=

Write left side as a fraction.

17 4

1 ± 4

1 The solutions are + 4

)( )

Find the square root of each side.

17 4

1 4

Add        to each side. 

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

Subtract c from each side.

bx –c x + a+ = a

Divide each side by a.

2

( )= a +( ) b –c + b x + = ( 2a ) a 4a

bx b x + + a 2a 2

2

–c

b 2a 2

2

2

(

b x+ 2a

)

2

– 4ac + b 2 = 2 4a

2

Add the square of half the coefficient  of x to each side. 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

–b ± b − 4ac 2

x=

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.