The graphs of \(f(x) = 7\) and \(g(x) = x^2 + 2x - 8\) intersect at \(x = -5\) and \(x = 3\). The correct solutions to the equation \(f(x) = g(x)\) are options a (-5) and c (3).
Let's first graph the functions \(f(x) = 7\) and \(g(x) = x^2 + 2x - 8\) on the same coordinate plane.
1. **Graph of \(f(x) = 7\):**
This is a horizontal line parallel to the x-axis at \(y = 7\).
2. **Graph of \(g(x) = x^2 + 2x - 8\):**
This is a quadratic function, and its graph is a parabola.
Now, let's find the solutions to the equation \(f(x) = g(x)\), which means we need to find the x-values where the two graphs intersect.
\[ 7 = x^2 + 2x - 8 \]
Combine like terms and set the equation to zero:
\[ x^2 + 2x - 15 = 0 \]
Now, factor the quadratic:
\[ (x - 3)(x + 5) = 0 \]
So, the solutions are \(x = 3\) and \(x = -5\).
Now, let's check the given options:
a. -5 (Correct)
b. -3 (Not a solution)
c. 3 (Correct)
d. 5 (Not a solution)
e. 7 (Not a solution)
Therefore, the correct solutions are \(x = -5\) and \(x = 3\), which correspond to options a and c.