To find the value of \( x \) in the equation \( f(x) = x^2 - 2x + 7 \) when \( f(x) = 10 \), we can set up the equation:
\[ x^2 - 2x + 7 = 10 \]
Now, we can solve for \( x \):
- Subtract 10 from both sides:
\[ x^2 - 2x + 7 - 10 = 0 \]
This simplifies to:
\[ x^2 - 2x - 3 = 0 \]
- Now, we can factor the quadratic equation:
\[ (x - 3)(x + 1) = 0 \]
- Setting each factor to zero gives us the possible solutions for \( x \):
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
Thus, the values of \( x \) that satisfy the equation \( f(x) = 10 \) are \( x = 3 \) and \( x = -1 \).