Question

Determine whether 9x^2 – 120x – 400 is a perfect square trinomial. If so, choose the correct factorization.

Answer 1

The quadratic expression 9x^2 − 120x − 400 is not a perfect square trinomial as it does not satisfy the conditions for being one. Thus, it cannot be factorized as a perfect square.

Step-by-step explanation:

The question asks us to determine whether the quadratic expression 9x^2 − 120x − 400 is a perfect square trinomial, and if so, to factorize it. A perfect square trinomial is an expression that can be written as the square of a binomial. For a trinomial ax2 + bx + c to be a perfect square, the second term b must be equal to 2 times the square root of ac, and c must be the square of half the coefficient of b.

Let's verify this for the given expression:

1. First, check if the constant term c is the square of half the coefficient of b:
2. For 9x2 − 120x − 400, b = − 120 and c = − 400.
3. Half of − 120 is − 60, and squaring it gives us 3600, which does not equal − 400. Therefore, the expression is not a perfect square trinomial.
4. Secondly, we see that b is not equal to two times the square root of ac, since 120 is not equal to 2 ∗ √(−9 ∗ −400), so the expression cannot be factorized as a perfect square.

Since both conditions for a perfect square trinomial are not met, 9x2 − 120x − 400 is not a perfect square trinomial and, therefore, cannot be factorized as such.

Answer 2

Given the expression

`9x^2-120x-400`

9 is a perfect square of

`9=(3)^2`

400 is also a perfect square of

`400=(20)^2`

Both the x-term and the constant are negative, which indicates that this polynomial is the result of a difference of squares:

`(a+b)(a-b)=a^2-ab+ab-b^2=a^2-b^2`

So the original factorization is

`(3x+20)(3x-20)`