Listen. Select the perfect square trinomial from among the options below.
A) [tex]x^2-12 x+36[/tex]
B) [tex]x^2+4 x-4[/tex]
C) [tex]x^2-2 x-3[/tex]
D) [tex]x^2+10 x+20[/tex]
Answer (2)
Check if the constant term is a perfect square.
Verify if the middle term is twice the product of the square roots of the first and last terms.
Option A, x 2 − 12 x + 36 , satisfies these conditions.
Therefore, the perfect square trinomial is x 2 − 12 x + 36 .
Explanation
Understanding the Problem We are given four trinomials and we need to identify the perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, i.e., ( a x + b ) 2 or ( a x − b ) 2 . The general form of a perfect square trinomial is a 2 x 2 + 2 ab x + b 2 or a 2 x 2 − 2 ab x + b 2 .
Solution Plan We need to check each option to see if it fits the form of a perfect square trinomial.
Checking Option A Option A: x 2 − 12 x + 36 . We check if the constant term is a perfect square and if the middle term is twice the product of the square roots of the first and last terms. Here, 36 = 6 2 and 12 x = 2 ⋅ x ⋅ 6 , so it is a perfect square trinomial: ( x − 6 ) 2 = x 2 − 12 x + 36 .
Checking Option B Option B: x 2 + 4 x − 4 . The constant term is -4, which is not a perfect square, so it is not a perfect square trinomial.
Checking Option C Option C: x 2 − 2 x − 3 . The constant term is -3, which is not a perfect square, so it is not a perfect square trinomial.
Checking Option D Option D: x 2 + 10 x + 20 . The constant term is 20, which is not a perfect square, so it is not a perfect square trinomial.
Conclusion Therefore, the perfect square trinomial is option A.
Examples
Perfect square trinomials are useful in various applications, such as completing the square to solve quadratic equations, simplifying algebraic expressions, and modeling physical phenomena. For example, in physics, the equation of motion for a simple harmonic oscillator can be expressed using a perfect square trinomial.
The perfect square trinomial from the given options is x 2 − 12 x + 36 (Option A), which can be expressed as ( x − 6 ) 2 . The other options do not meet the criteria for perfect squares.
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