- Set the function \(f(x) = x^2 + 5x - 36\) equal to 0.
- Factor the expression as \((x - 4)(x + 9) = 0\).
- Find the roots \(x = 4\) and \(x = -9\).
- State the final \(x\)-intercepts as \((4, 0)\) and \((-9, 0)\).
- The final answer is \(\boxed{(4, 0) \text{ and } (-9, 0)}\).
Explanation
- Analyzing the Problem
To find the \(x\)-intercepts of the graph of the function \(f(x) = x^2 + 5x - 36\), we must determine the values of \(x\) for which \(f(x) = 0\). This requires solving the quadratic equation:
\[x^2 + 5x - 36 = 0\]
We are looking for the points \((x, 0)\) that lie on the graph of the function.
- Factoring the Quadratic Equation
To solve the quadratic equation \(x^2 + 5x - 36 = 0\), we look for two numbers whose product is \(-36\) and whose sum is \(5\).
Testing factors of \(-36\):
- \(-1 \times 36 = -36\), \(-1 + 36 = 35\)
- \(-2 \times 18 = -36\), \(-2 + 18 = 16\)
- \(-3 \times 12 = -36\), \(-3 + 12 = 9\)
- \(-4 \times 9 = -36\), \(-4 + 9 = 5\)
Since the numbers \(-4\) and \(9\) satisfy these conditions, we can factor the quadratic expression as:
\[(x - 4)(x + 9) = 0\]
- Identifying the x-intercepts
Setting each factor to zero gives us:
1) \(x - 4 = 0 \implies x = 4\)
2) \(x + 9 = 0 \implies x = -9\)
Therefore, the function \(f(x)\) equals zero when \(x = 4\) or \(x = -9\). This corresponds to the \(x\)-intercepts \((4, 0)\) and \((-9, 0)\). Comparing this to the given choices, we select \((4, 0)\) and \((-9, 0)\).
Examples
Understanding x-intercepts is vital in physics and engineering. For example, if you throw a ball, the trajectory can be described by a quadratic equation where the x-axis represents time. The x-intercepts tell you the time when the ball leaves the ground and the time when it hits the ground again. Solving such equations allows you to predict motion accurately.