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)
A perfect square trinomial has the form a 2 + 2 ab + b 2 or a 2 − 2 ab + b 2 .
Check each option to see if it fits the form.
Option A, x 2 − 12 x + 36 , can be written as ( x − 6 ) 2 .
Therefore, the perfect square trinomial is x 2 − 12 x + 36 .
Explanation
Understanding Perfect Square Trinomials We need to identify the perfect square trinomial from the given options. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is a 2 + 2 ab + b 2 or a 2 − 2 ab + b 2 .
Checking Each Option Let's examine each option:
Option A: x 2 − 12 x + 36 This can be rewritten as x 2 − 2 ( x ) ( 6 ) + 6 2 . This fits the form a 2 − 2 ab + b 2 , where a = x and b = 6 . Thus, it can be factored as ( x − 6 ) 2 .
Option B: x 2 + 4 x − 4 For this to be a perfect square trinomial, the constant term must be positive. However, − 4 is negative, so this is not a perfect square trinomial.
Option C: x 2 − 2 x − 3 Similar to option B, the constant term is negative ( − 3 ), so this is not a perfect square trinomial.
Option D: x 2 + 10 x + 20 For this to be a perfect square trinomial, the constant term must be a perfect square. Half the coefficient of the x term is 10/2 = 5 . 5 2 = 25 , but the constant term is 20 , so this is not a perfect square trinomial.
Conclusion Therefore, option A, x 2 − 12 x + 36 , is the perfect square trinomial.
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. Understanding and identifying perfect square trinomials helps in simplifying and solving such problems efficiently.
The perfect square trinomial among the given options is A) x 2 − 12 x + 36 . This expression can be factored as ( x − 6 ) 2 .
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