Let's analyze the given equation [tex]\((x-4)(x+2) = 16\)[/tex] step-by-step to determine which of the provided statements is true.
1. Expanding and simplifying the equation:
Start with the given equation and expand the left-hand side.
[tex]\[
(x-4)(x+2) = 16
\][/tex]
When we expand [tex]\((x-4)(x+2)\)[/tex], we get:
[tex]\[
x^2 + 2x - 4x - 8 = x^2 - 2x - 8
\][/tex]
So the equation becomes:
[tex]\[
x^2 - 2x - 8 = 16
\][/tex]
2. Converting to standard form:
To convert this into a standard quadratic equation, we need to bring all terms to one side of the equation:
[tex]\[
x^2 - 2x - 8 - 16 = 0
\][/tex]
Simplify the left-hand side:
[tex]\[
x^2 - 2x - 24 = 0
\][/tex]
3. Examining each statement:
(a) The equation [tex]\(x-4=16\)[/tex] can be used to solve for a solution of the given equation.
No, this is not correct. The equation [tex]\(x - 4 = 16\)[/tex] does not directly relate to the given factored form of [tex]\((x-4)(x+2)=16\)[/tex].
(b) The standard form of the equation is [tex]\(x^2 - 2 x - 8 = 0\)[/tex].
No, the standard form derived from the equation [tex]\((x-4)(x+2) = 16\)[/tex] is [tex]\(x^2 - 2x - 24 = 0\)[/tex], not [tex]\(x^2 - 2x - 8 = 0\)[/tex].
(c) The factored form of the equation is [tex]\((x+4)(x-6)=0\)[/tex].
No, without solving the quadratic equation [tex]\(x^2 - 2x - 24 = 0\)[/tex], we cannot determine the factored form directly like [tex]\((x+4)(x-6)=0\)[/tex]. More importantly, we have not derived such a factorization.
(d) One solution of the equation is [tex]\(x = -6\)[/tex].
While we have not solved the equation [tex]\(x^2 - 2x - 24 = 0\)[/tex] to find the roots, the standard form transformation itself doesn't give us the roots directly. We need to solve the quadratic equation first to verify.
Given our detailed steps and analysis:
- By expanding the left side of the given equation, the correct standard form is indeed [tex]\(x^2 - 2 x - 24 = 0\)[/tex].
Thus, none of the provided statements are true based on the derived standard form and our mathematical derivations from the given equation [tex]\((x-4)(x+2) = 16\)[/tex].
