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Concept Version 15
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Factoring Perfect Square Trinomials

When a trinomial is a perfect square, it can be factored into two equal binomials.

Learning Objective

  • Evaluate whether a quadratic equation is a perfect square and factor it accordingly if it is


Key Points

    • Some quadratics, known as perfect squares, can be factored into two equal binomials.
    • Perfect squares have the form a2+2ab+b2a^2+2ab+b^2a​2​​+2ab+b​2​​.
    • Perfect square quadratics have only one root.

Terms

  • trinomial

    A polynomial expression consisting of three terms, or monomials, separated by addition and/or subtraction symbols.

  • binomial

    A polynomial consisting of two terms, or monomials, separated by addition or subtraction symbols.


Full Text

Recognizing Perfect Square Trinomials

Note that if a binomial of the form a+ba+ba+b is squared, the result has the following form: (a+b)2=(a+b)(a+b)=a2+2ab+b2.(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.(a+b)​2​​=(a+b)(a+b)=a​2​​+2ab+b​2​​. So both the first and last term are squares, and the middle term has factors of 2,2, 2, aaa, and b,b,b, where the latter are the square roots of the first and last term respectively. 

For example, if the expression 2x+32x+32x+3 were squared, we would obtain (2x+3)(2x+3)=4x2+12x+9.(2x+3)(2x+3)=4x^2+12x+9.(2x+3)(2x+3)=4x​2​​+12x+9. Note that the first term 4x24x^24x​2​​ is the square of 2x2x2x while the last term 999 is the square of 333, while the middle term is twice 2x⋅32x\cdot32x⋅3.

It is important to be able to recognize such trinomials, so that they can the be factored as a perfect square. 

Factoring Perfect Square Trinomials

If you are attempting to to factor a trinomial and realize that it is a perfect square, the factoring becomes much easier to do. 

Example 1

Suppose you were trying to factor x2+8x+16.x^2+8x+16.x​2​​+8x+16. One can see that the first term is the square of xxx while the last term is the square of 444. Since the middle term is twice 4⋅x4 \cdot x4⋅x, this must be a perfect square trinomial, and we can factor it as: 

x2+8x+16=(x+4)2x^2+8x+16=(x+4)^2x​2​​+8x+16=(x+4)​2​​

Example 2

Suppose you were trying to solve 9x2+6x+1=0.9x^2+6x+1=0.9x​2​​+6x+1=0. You might try to factor the quadratic expression on the left-hand side of the equation. Since the first term is 3x3x3x squared, the last term is one squared, and the middle term is twice 3x⋅13x\cdot 13x⋅1, this is a perfect square, and we can write:

9x2+6x+1=(3x+1)29x^2+6x+1=(3x+1)^29x​2​​+6x+1=(3x+1)​2​​ 

Thus the original equation has the form (3x+1)2=0(3x+1)^2=0(3x+1)​2​​=0, so the only solution is when 3x+1=03x+1=03x+1=0, which is when 3x=−1,3x=-1,3x=−1, or x=−13.x=-\frac{1}{3}. x=−​3​​1​​. This quadratic equation has only that one solution.

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