How do you find the x-intercepts of the function f(x) = x² + 4x – 12?

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Finding the x-Intercepts

To find the x-intercepts of the function f(x) = x² + 4x – 12, we need to determine the values of x where the graph of the function crosses the x-axis. This occurs where the function equals zero, so we set up the equation:

x² + 4x – 12 = 0

Step 1: Factoring the Quadratic Equation

We can factor the quadratic equation. We are looking for two numbers that multiply to -12 (the constant term) and add to 4 (the coefficient of the linear term). Those numbers are 6 and -2.

Thus, we can rewrite the equation as:

(x + 6)(x – 2) = 0

Step 2: Solving for x

Next, we set each factor equal to zero:

  • x + 6 = 0 ⟹ x = -6
  • x – 2 = 0 ⟹ x = 2

Conclusion

The x-intercepts of the function f(x) = x² + 4x – 12 are x = -6 and x = 2. This means the graph intersects the x-axis at the points (-6, 0) and (2, 0).

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