To solve the quadratic equation [tex]\( f(x) = x^2 + 2x - 15 = 0 \)[/tex] using its graph, we need to identify the points where the graph crosses the x-axis. These points correspond to the solutions or roots of the equation.
1. Understanding the Quadratic Function and its Graph:
- The given quadratic function is [tex]\( f(x) = x^2 + 2x - 15 \)[/tex].
- The graph of this function is a parabola that opens upwards because the coefficient of [tex]\( x^2 \)[/tex] is positive.
2. Identifying the Roots:
- The roots of the quadratic equation are the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
- On the graph, these roots are the x-coordinates of the points where the parabola intersects the x-axis.
3. Finding the Intersection Points:
- By analyzing the graph, we find that the parabola intersects the x-axis at two points. These points are located at [tex]\( x = -5 \)[/tex] and [tex]\( x = 3 \)[/tex].
4. Verifying the Solutions from the Graph:
- The x-coordinates of the intersection points must satisfy the original equation [tex]\( f(x) = x^2 + 2x - 15 \)[/tex].
- Substituting [tex]\( x = -5 \)[/tex] into the equation:
[tex]\[ f(-5) = (-5)^2 + 2(-5) - 15 = 25 - 10 - 15 = 0 \][/tex]
- Substituting [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[ f(3) = (3)^2 + 2(3) - 15 = 9 + 6 - 15 = 0 \][/tex]
From the graph, the correct pair of solutions where [tex]\( f(x) = 0 \)[/tex] is:
[tex]\[ x = -5 \text{ or } x = 3 \][/tex]
Thus, the correct answer from the given options is:
[tex]\[ \boxed{x = -5 \text{ or } x = 3} \][/tex]
