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Select the correct answer.

What are the [tex]$x$[/tex]-intercepts of this quadratic function?
[tex]\[ g(x) = -2(x-4)(x+1) \][/tex]

A. [tex]\((4,0)\)[/tex] and [tex]\((1,0)\)[/tex]

B. [tex]\((-4,0)\)[/tex] and [tex]\((1,0)\)[/tex]

C. [tex]\((4,0)\)[/tex] and [tex]\((-1,0)\)[/tex]

D. [tex]\((-4,0)\)[/tex] and [tex]\((-1,0)\)[/tex]


Sagot :

To determine the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(g(x) = -2(x-4)(x+1)\)[/tex], we need to find the points where the function crosses the [tex]\(x\)[/tex]-axis. This occurs when [tex]\(g(x) = 0\)[/tex].

Setting the function equal to zero, we have:
[tex]\[ -2(x-4)(x+1) = 0 \][/tex]

Next, we solve for the values of [tex]\(x\)[/tex] that make this equation true. We can use the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must be zero.

We have two factors in our equation: [tex]\((x-4)\)[/tex] and [tex]\((x+1)\)[/tex]. We set each factor equal to zero and solve for [tex]\(x\)[/tex].

1. Set [tex]\(x-4=0\)[/tex]:
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]

2. Set [tex]\(x+1=0\)[/tex]:
[tex]\[ x + 1 = 0 \implies x = -1 \][/tex]

Thus, the [tex]\(x\)[/tex]-intercepts of the function are the points [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].

Therefore, the correct answer is:
C. [tex]\((4,0)\)[/tex] and [tex]\((-1,0)\)[/tex]