To find the values of [tex]\( x \)[/tex] such that the function [tex]\( f(x) = -2x^2 + 4x + m \)[/tex] has the same value as [tex]\( f(-2) \)[/tex], we follow these steps:
1. Substitute [tex]\( x = -2 \)[/tex] into the function to find [tex]\( f(-2) \)[/tex]:
[tex]\[
f(-2) = -2(-2)^2 + 4(-2) + m
\][/tex]
Simplify the expression:
[tex]\[
f(-2) = -2(4) + 4(-2) + m = -8 - 8 + m = m - 16
\][/tex]
2. Set up the equation [tex]\( f(x) = f(-2) \)[/tex]. We know [tex]\( f(-2) = m - 16 \)[/tex], so we set:
[tex]\[
-2x^2 + 4x + m = m - 16
\][/tex]
3. Eliminate [tex]\( m \)[/tex] from both sides of the equation:
[tex]\[
-2x^2 + 4x + m - m = m - 16 - m
\][/tex]
This simplifies to:
[tex]\[
-2x^2 + 4x = -16
\][/tex]
4. Rearrange the equation to standard form:
[tex]\[
-2x^2 + 4x + 16 = 0
\][/tex]
5. Factor or use the quadratic formula [tex]\( ax^2 + bx + c = 0 \)[/tex] to solve for [tex]\( x \)[/tex]. In this case, it's easier to factor:
Let's divide everything by -2 to simplify:
[tex]\[
x^2 - 2x - 8 = 0
\][/tex]
Now, factor the quadratic equation:
[tex]\[
(x - 4)(x + 2) = 0
\][/tex]
6. Solve for the values of [tex]\( x \)[/tex]:
Setting each factor equal to zero gives us the solutions:
[tex]\[
x - 4 = 0 \quad \text{or} \quad x + 2 = 0
\][/tex]
Therefore:
[tex]\[
x = 4 \quad \text{or} \quad x = -2
\][/tex]
Hence, the function [tex]\( f(x) \)[/tex] has the same value as [tex]\( f(-2) \)[/tex] for the values [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
