Download Completing the Square

PERFECT SQUARE TRINOMIALS
Any trinomial of the form ax2 + bx + c that can be
factored to be a (BINOMIAL Factor) squared
Sum Factors:
Difference Factors:
a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2
(1) 9x2 + 12x + 4
(2) x2 - 8x + 16
(3) 4x2 - 20x + 25
(4) x2 + 20x + 100
How do you make a perfect square trinomial?
• STEP 1: DIVIDE middle term value (b-value) by 2
• STEP 2: SQUARE it
• STEP 3: Make your step 2 answer the constant
FACTORS:
Binomial is add if middle term is positive
Binomial is subtract if middle term is negative
EXAMPLE: x2 + 6x + c
EXAMPLE: x2 - 10x + c
Middle term: 6
Middle term: -10
Divide by 2: 3
Divide by 2: -5
Squared = 9
Squared = 25
x2 + 6x + 9 = (x + 3)2
x2 – 10x + 25 = (x - 5)2
Create Perfect Square Trinomials
Practice finding “c”
• x2 - 8x + c
• x2 - 3x + c
• x2 + 10x + c
• x2 + 9x + c
Continued: Practice finding “c”
1
3
5
2
2
2
x  xc
x  xc
x  xc
2
4
6
STEPS for COMPLETING THE SQUARE
ax2 + bx + c = 0
Step 1: Lead coefficient of x2 must be 1
• DIVIDE by “a” value
Step 2: Subtract current ‘c’ term
Step 3: Find value to make a perfect square trinomial
• Divide middle term, “bx”, by 2 and square
• Add that value to both sides of equation
Step 4: Factor (perfect square!)
*Shortcut = half of middle term is part of binomial factor*
Step 5: Solve for x
Example: Solve by completing the square
• x2 + 6x + 4 = 0
- SUBTRACT 4
• x2 + 6x = - 4
-Find the constant value to
create a perfect square and
ADD to both sides
(half of 6 is 3, 3 squared is 9)
• x2 + 6x + 9 = -4 + 9 -FACTOR perfect square
trinomial
• (x + 3)2 = 5
x3  5
x  3  5
-SOLVE for x:
Square root both sides
Use plus or minus
(Check to simplify radical)
Practice #1: Completing the Square
2
2
x

4
x

11

0
2. x  12 x  13  0
1.
3. x  5 x  2  0
2
4.
3
x  7x   0
4
2
Example with leading coefficient
2
- Divide every number by 2
2x  4x  3  0
3
x  2x   0
2
3
2
x  2x 
2
3
2
x  2x  1   1
2
2
5
( x  1) 
2
2
- Add 3/2 on both sides
- Find c to make perfect
square trinomial
(half of 2 = 1, 1 squares = 1
- Factor left side, combine
like terms on the right
- Solve for x:
Square Root with plus/minus
Rationalize Fraction Radicals
1.
Practice #2: Completing the Square
2
2
2.
3x  9x  4  0
2 x  11 x  3  0
3.
2x  8x  1  0
2
4.
5 x  12 x  2  0
2
Practice: Equations with Complex Solutions
1. x 2  4 x  8  0
3.
x  6 x  10  0
2
2
2
x
 4x  3  0
2.
4. 4 x 2  6 x  5  0
Practice : Solve Equations to equal zero?
2
2
1. x  4 x  8  6
2. 2 x  4 x  3  3 x  5
3.
x  8x  2  2x
2
4.
2 x  3 x  7  1  9 x
2