find the x-intercept: \( y=x^{2}-4 x-5 \)

To find the **x-intercepts** of the quadratic function \( y = x^{2} - 4x - 5 \), you need to determine the points where the graph of the function crosses the x-axis. At these points, the value of \( y \) is **0**. ### Step 1: Set \( y \) to 0 \[ 0 = x^{2} - 4x - 5 \] ### Step 2: Solve the Quadratic Equation You can solve the quadratic equation \( x^{2} - 4x - 5 = 0 \) by using the **quadratic formula**: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] For the given equation: - \( a = 1 \) - \( b = -4 \) - \( c = -5 \) Plugging these values into the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{(-4)^{2} - 4(1)(-5)}}{2(1)} \] \[ x = \frac{4 \pm \sqrt{16 + 20}}{2} \] \[ x = \frac{4 \pm \sqrt{36}}{2} \] \[ x = \frac{4 \pm 6}{2} \] This gives two solutions: 1. **First solution:** \[ x = \frac{4 + 6}{2} = \frac{10}{2} = 5 \] 2. **Second solution:** \[ x = \frac{4 - 6}{2} = \frac{-2}{2} = -1 \] ### Step 3: State the X-Intercepts The x-intercepts are the points where \( y = 0 \), so: - **First x-intercept:** \( (5, 0) \) - **Second x-intercept:** \( (-1, 0) \) ### **Conclusion** The quadratic function \( y = x^{2} - 4x - 5 \) intersects the x-axis at the points **(5, 0)** and **(−1, 0)**.