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

To find the x-intercept of the equation \( y = x^2 - 4x - 5 \), set \( y \) to 0 and solve for \( x \). So, the equation becomes \( 0 = x^2 - 4x - 5 \). Factoring gives us \( (x - 5)(x + 1) = 0 \). This means \( x - 5 = 0 \) or \( x + 1 = 0 \). Thus, the x-intercepts are \( x = 5 \) and \( x = -1 \). Now, for a fun nugget of trivia: this quadratic function actually represents the shape of a parabola! In this case, it opens upwards because the coefficient of \( x^2 \) is positive. Imagine this parabola gracefully curving up, with its hands reaching down to touch the x-axis at those beautiful points (5,0) and (-1,0). You can think of the x-intercepts as the "landing spots" where the graph touches the x-axis, and they mark the points where the output (y) is zero. These intercepts play a crucial role in graphing and analyzing polynomial functions, giving insight into the function's behavior.