Factor a Perfect Square Trinomial
High School Algebra
Aligned to Common Core State Standards
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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
Taking this in the 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
(7x + 2)2
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.
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
Addition Property of Equality
Addition Property of Equality
Distributive Property
Zero Product Property
Addition Property of Equality
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
Addition Property of Equality
(x – 3)2 = 0
Distributive Property
x–3=0 x–3=0
Zero Product Property
x=3
Addition Property of Equality
x=3
Addition Property of Equality
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
The graph will intersect the x-axis at (.4, 0).
It will open up since the coefficient of x2 is positive. The axis
of symmetry is x = –b/2a or x = –(-20)/2(25) = 20/50 = 2/5.
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