Question

Find the x-intercepts of the graph of the equation. $$y=-x^{2}+4 x+1$$

Short answer

The x-intercepts of the function are \(x = 2 + \sqrt{5}\) and \(x = 2 - \sqrt{5}\).

Step-by-step solution

Set the Function to Zero

Set the function \(y = -x^2 + 4x + 1\) equal to zero to find the x-values when y = 0, i.e. \(0 = -x^2 + 4x + 1\).

Rearrange the Equation

Rearrange the equation to the standard quadratic form, i.e. \(x^2 - 4x - 1 = 0.\)

Solving the Quadratic Equation

Use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for a quadratic equation in the form \(ax^2 + bx + c = 0\). In this case, \(a = 1\), \(b = -4\), and \(c = -1\). The solutions to the equation are \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4*1*(-1)}}{2*1} = 2 \pm \sqrt{5}\).

Check the Solutions

Substitute \(x = 2 + \sqrt{5}\) and \(x = 2 - \sqrt{5}\) into the equation and confirm that they satisfy \(y = -x^2 + 4x + 1\).