Question

The function g is given in three equivalent forms. Which form most quickly reveals the zeros (or "roots") of the function?

A g(x)=-2(x+4)^2+18
B g(x)=-2x^2-16x-14
C g(x)=-2(x+1)(x+7)
D None of these

Answer 1

The form that most quickly reveals the zeros of the function is C, g(x) = -2(x+1)(x+7). So, Option C is correct.

Step-by-step explanation:

In order to identify the form that most quickly reveals the zeros of the function, we should look for a factored form that directly provides information about the roots. The factored form of a quadratic equation, such as C, g(x) = -2(x+1)(x+7), explicitly displays the roots. In this case, the roots are given by x = -1 and x = -7. The factored form allows for a straightforward identification of the zeros without the need for additional algebraic manipulations.

Let's briefly explore the other options to emphasize why they are not as efficient in revealing the zeros. Option A, g(x) = -2(x+4)^2+18, is an expanded and shifted form, making it less obvious to identify the roots without additional computation. Option B, g(x) = -2x^2-16x-14, is an expanded form that requires quadratic formula or factoring to find the zeros. Therefore, the factored form in option C is the most efficient, as it directly provides the zeros without the need for extra steps.

In summary, option C, g(x) = -2(x+1)(x+7), is the most effective form for quickly revealing the zeros of the function because it presents the roots explicitly in factored form, eliminating the need for additional calculations.