Transcript Lesson 8-8 - Elgin Local Schools

Lesson 9-6
Perfect Squares and Factoring
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
• Questions to ask.
•16x2 + 32x + 64
•Is the first term a perfect square?
Yes, 16x2 = (4x)2
•Is the last term a perfect square?
Yes, 64 = 82
•Is the middle term equal to 2(4x)(8)?
No, 32x  2(4x)(8)
16x2 + 32x + 64 is not a perfect square trinomial.
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
• 25x2 - 30x + 9
•49x2 + 42x + 36
Number of Terms
Factoring Technique
2 or more
Greatest Common Factor
2
Difference of Squares
Perfect square trinomial
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
x2 + bx + c
x2 + bx + c = (x + m)(x + n),
when m + n = b and mn =
c.
ax2 + bx + c
ax2 + bx + c = ax2 + mx +
nx+ c, when m + n = b and
mn = ac. Then using
factoring by grouping
Factoring by grouping
ax + bx + ay + by
= x(a + b) + y(a + b)
= (a + b)(x + y)
3
4 or more
a2 - b2 = (a + b)(a - b)
Ex. 2
Factor Completely
Factor each polynomial
•4x2 - 36
•First check for the GCF. Then, since the polynomial has two terms, check for
the difference of two squares.
4x2 - 36
= 4(x2- 9)
4 is the GCF
= 4(x2 - 32)
x2 = x  x, and 9 = 3 3
= 4(x + 3)(x - 3)
factor the difference of squares.
Ex. 2
Factor Completely
25x2 + 5x - 6
•This polynomial has three terms that have a GCF of 1. While the first term is
a perfect square 25x2 = (5x)2, the last term is not. Therefore, this is not a
perfect square trinomial.
•This trinomial is one of the form ax2 + bx + c. Are there two numbers m and n
whose product is 25  -6 or -150 and whose sum is 5? Yes, the product of 15
and -10 is -150 and their sum is 5.
25x2 + 5x - 6
= 25x2 + mx + nx - 6
Write the pattern
= 25x2 + 15x - 10x - 6
m = 15 and n = -10
= (25x2 + 15x) + ( -10x - 6)
Group terms with common factors.
= 5x(5x + 3) -2 ( 5x + 3)
Factor out the GCF for each grouping.
= (5x + 3) (5x - 2)
5x + 3 is the common factor.
Factor each polynomial.
•6x2 - 96
•16x2 + 8x -15
Ex. 3
Solve Equations with Repeated Factors.
Solve x2 - x + ¼ = 0
x2 - x + ¼ = 0
Original equation
x2 - 2(x)(½) + (½)2 = 0
Recognize x2 - x + ¼ as a perfect square trinomial
(x - ½)2 = 0
Factor the perfect square trinomial.
x-½=0
Set repeated factor equal to zero.
x=½
Solve for x.
Factor each polynomial.
•4x2 + 36x + 81
Key Concept
• For any number n > 0, if x2 = n, then x =  n
x2 = 9
x2
=  9 or  3
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Ex. 4
Use the Square Root Property to Solve Equations
Solve (a + 4)2 = 49
(a + 4)2 = 49
a+4=
49
Original equation
Square Root Property
a+4=7
49 = 7  7
a = -4  7
Subtract 4 from each side.
a = -4 + 7 or a = -4 - 7
Separate into two equations.
a = 3 or a = -11
Simplify
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Ex. 4
Use the Square Root Property to Solve Equations
Solve y2 -4y + 4 = 25
y2 -4y + 4 = 25
Original equation
(y)2 -2(y)(2) + 22
Recognize perfect square trinomials.
(y - 2)2 = 25
Factor perfect square trinomial.
y-2=
Square root property.
25
y-2=5
25 = 5  5
y = 2 + 5 or y = 2 - 5
Separate into two equations.
y = 7 or y = -3
Simplify
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Ex. 4
Use the Square Root Property to Solve Equations
Solve (x - 3)2 = 5
(x - 3)2 = 5
x-3=
x =3
Original equation
5
5
Square root property.
Add 3 to each side.
x = 3 + 5 or x = 3 - 5
Separate into two equations.
x ≈ 5.24 or x ≈ 0.76
Simplify
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Solve each equation. Check your solutions.
•(b - 7)2 = 36
•y2 + 12y + 36 = 100
•(x + 9)2 = 8