Transcript 5.4 Factoring Quadratic Equations

5-4 Factoring Quadratic
Expressions
Perfect square trinomials
What is a perfect square trinomial?
 A perfect square trinomial is the product you get when
you square a binomial (you multiply the binomial by
itself)
 To square a binomial
 Example:
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
(x+5) (x+5)
square the first terms
_____ x _____= _____
square the last terms
_____ x _____= _____
the middle term is two times the product of the terms
2( _____ _____) = ______
The result is x2 + 10x + 25
 Factoring a perfect square trinomial when
all terms are positive
 9x2 + 48x + 64
 Take the square root of the first and third terms

First term (3x)
third term (8)
 Put the terms in the parenthesis squared and separate
the terms with a plus sign.

(3x + 8)2
 4x2 + 24x + 36
 Take the square root of the first and third terms and
separate them by an addition sign
 (2x + 6)2
 Check your answer:



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Square the first term (2x)2 = 4x2
Square the last term (6)2 = 36
Multiply the two terms together and double them
(2(2x)(6)) =24x
Try this one:
4x2 + 28x + 49
 Factoring a perfect square trinomial when
the middle term is negative
 4n2 – 20n + 25
 The only difference with factoring these trinomials is
that the sign between the two square roots is negative
 Take the square root of the first and third terms

First term (2n)
third term (5)
 Separate the terms with a negative sign
 (2n - 5)2
 4n2 – 16n + 16
 Take the square root of the first and third terms and
separate the numbers with a subtraction sign

(2n - 4)2
 Check your answer:
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Square the first term (2n)2 = 4n2
Square the last term (4)2 = 16
Multiply the two terms together and double them
(2(2n)(-4)) =-16n
Try this one:
9x2 - 42x + 49
Factoring the difference of two squares
 The difference of two squares is written as:
a 2 – b2
 When they are factored they become (a+b)(a-b)
 When the terms of the binomials are FOIL’d the
middle terms cancel each other out because one is
positive and one is negative
 (a+b)(a-b) = a2 + ab – ab - b2 = a2 – b2
 Take the square root of the first term and the square
root of the second term and place them into two sets
of parentheses – one set separated with a plus sign and
one set separated with a minus sign
 Example:
c2 – 64 = (c+8)(c-8)
 Square root of the first term is c
 Square root of the last term is 8
 Put the terms into two parentheses and separate one
with a plus sign and one with a minus sign
 Remember: the middle terms will cancel out when the
binomials of the difference of two squares are
multiplied together.
 Let’s try this one:
4x2 – 16
 Take the square root of the first term: 2x
 Take the square root of the last term:
4
 Rewrite the terms in two parentheses
 (2x 4) (2x 4)
 Separate the terms by a plus sign in one of the
parentheses and a minus sign in the other parentheses
(2x + 4) (2x – 4)
 How about this one:
9x2 – 36
 One last note:
 Sometimes you may have to factor out the GCF before
you can factor the quadratic.
 You can try to find factors of the first term and then find
factors of the last term to make two binomials that you
can multiply together
 If you can factor out a GCF first – do so in order to make
the factoring easier
 3n2 – 24n – 27
 3(n2 – 8n – 9)
 3(n – 9)(n + 1)
 This technique works for any trinomial you are trying to
factor
h/w:
p. 264: 38, 39, 41, 42, 44, 45,
52, 55, 57, 58