To find the x-intercepts of the function f(x) = x² + 5x – 36, we need to set the function equal to zero and solve for x:
f(x) = 0
x² + 5x – 36 = 0
This is a quadratic equation, and we can solve it using the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, a = 1, b = 5, and c = -36. Plugging these values into the quadratic formula gives:
x = (−5 ± √(5² – 4 * 1 * (−36))) / (2 * 1)
First, calculate the discriminant (the part under the square root):
5² – 4 * 1 * (−36) = 25 + 144 = 169
Now substituting back, we have:
x = (−5 ± √169) / 2
x = (−5 ± 13) / 2
This gives us two possible solutions:
1. x = (−5 + 13) / 2 = 8 / 2 = 4
2. x = (−5 – 13) / 2 = −18 / 2 = −9
So, the x-intercepts of the graph are at x = 4 and x = −9. Graphically, these points are where the parabola crosses the x-axis.