Alright, let's solve the equation [tex]\( f(x) = (x - 2)^2 - 25 \)[/tex] step by step.
1. Understand the function:
The function given is [tex]\( f(x) = (x - 2)^2 - 25 \)[/tex]. We need to find the roots of this function, which are the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
2. Set the function to zero:
We want to solve for [tex]\( x \)[/tex] where:
[tex]\[
(x - 2)^2 - 25 = 0
\][/tex]
3. Isolate the squared term:
Add 25 to both sides of the equation to isolate the squared term:
[tex]\[
(x - 2)^2 - 25 + 25 = 0 + 25
\][/tex]
Simplifying this, we get:
[tex]\[
(x - 2)^2 = 25
\][/tex]
4. Take the square root of both sides:
Taking the square root of both sides to solve for [tex]\( x - 2 \)[/tex]:
[tex]\[
\sqrt{(x - 2)^2} = \sqrt{25}
\][/tex]
This simplifies to:
[tex]\[
|x - 2| = 5
\][/tex]
5. Solve the absolute value equation:
The absolute value equation [tex]\( |x - 2| = 5 \)[/tex] gives us two possible linear equations:
[tex]\[
x - 2 = 5 \quad \text{or} \quad x - 2 = -5
\][/tex]
6. Solve each equation separately:
For the equation [tex]\( x - 2 = 5 \)[/tex]:
[tex]\[
x = 5 + 2
\][/tex]
[tex]\[
x = 7
\][/tex]
For the equation [tex]\( x - 2 = -5 \)[/tex]:
[tex]\[
x = -5 + 2
\][/tex]
[tex]\[
x = -3
\][/tex]
7. Conclusion:
The solutions to the equation [tex]\( f(x) = (x - 2)^2 - 25 = 0 \)[/tex] are:
[tex]\[
x = -3 \quad \text{and} \quad x = 7
\][/tex]
So, the roots of the function are [tex]\( x = -3 \)[/tex] and [tex]\( x = 7 \)[/tex].
