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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]
To determine the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(g(x) = -2(x-4)(x+1)\)[/tex], we need to solve for the values of [tex]\(x\)[/tex] that make [tex]\(g(x) = 0\)[/tex].
1. Set the function equal to zero: [tex]\[
-2(x - 4)(x + 1) = 0
\][/tex]
2. Factor the equation: [tex]\[
-2(x - 4)(x + 1) = 0
\][/tex] Since the product of two factors is zero, one or both of the factors must be zero.
3. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
- For the factor [tex]\((x - 4) = 0\)[/tex]: [tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
- For the factor [tex]\((x + 1) = 0\)[/tex]: [tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
4. Determine the [tex]\(x\)[/tex]-intercepts: The [tex]\(x\)[/tex]-intercepts occur at the points where the function crosses the [tex]\(x\)[/tex]-axis, which are the points [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
5. Select the correct option: From the given options, the correct answer is:
B. [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex]
Thus, the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(g(x) = -2(x-4)(x+1)\)[/tex] are [tex]\((4, 0)\)[/tex] and [tex]\((-1, 0)\)[/tex].
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