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Factor a Perfect Square Trinomial
High School Algebra
Aligned to Common Core State Standards
Teacher Notes
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Perfect Square Trinomials
Perfect square trinomials are the result of squaring a binomial.
(x + 9)2 = (x + 9)(x + 9) = x2 + 9x + 9x + 81 = x2 + 18x + 81
In general terms:
(a + b)2 = a2 + ab + ab + b2 = a2 + 2ab + b2
(a – b)2 = a2 – ab – ab + b2 = a2 – 2ab + b2
Expand the following.
(5t – 3)2 =
(w + 6)2 =
(2s – 7)2 =
(11x + 8)2 =
Expand the following.
(5t – 3)2 = 25t2 – 30t +9
(w + 6)2 = w2 + 12w + 36
(2s – 7)2 = 4s2 – 28s + 49
(11x + 8)2 = 121x2 + 176x + 64
Factoring a Perfect Square Trinomial
In general terms:
(a + b) 2 = a 2 + ab + ab + b2 = a2 + 2ab + b2
(a – b) 2 = a 2 – ab – ab + b2 = a2 – 2ab + b2
In reverse, you would factor a perfect square trinomial like this:
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Factoring Steps
Factor
49x2 + 28 x + 4
Take the square root of the first term,
the sign of the second term,
and the square root of the third term.
Square the quantity.
(7x + 2)2
Check the middle term by multiplying the first term and last term and
doubling.
[(7x)(2)]2 = 28x
This matches the middle term and therefore
is the correct factorization.
Factor the following.
m2 – 10m + 25 =
144p2 – 24p + 1 =
81n2 + 54n + 9 =
Factor the following.
m2 – 10m + 25 = (m – 5)2
144p2 – 24p + 1 = (12p – 1)2
81n2 + 54n + 9 = (9n + 3)2
Solve using the Zero Product Property.
Use the reasons given for each step to guide you.
x2 = 3(2x – 3)
x2 = 3(2x – 3)
Given
Distributive Property
Distributive Property
Zero Product Property
Solve using the Zero Product Property.
Use the reasons given for each step to guide you.
x2 = 3(2x – 3)
x2 = 3(2x – 3)
Given
x2 = 6x – 9
Distributive Property
x2 – 6x = - 9
x2 – 6x + 9 = 0
(x – 3)2 = 0
Distributive Property
x–3=0
x–3=0
Zero Product Property
x=3
x=3
Find the zeros of the function.
4x2  12x + 9 = 0
Solve and graph the solution.
25x2 + 4 = 20x
Solve and graph the solution.
25x2 + 4 = 20x
25x2  20x + 4 = 0
(5x  2)2 = 0
5x  2 = 0 5x  2 = 0
5x = 2
5x = 2
x = 2/5
x = 2/5
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