Given the roots of the quadratic are \( 1 \) and \( 5 \), write the quadratic in standard form.

To find the quadratic in standard form given the roots \( 1 \) and \( 5 \), you can start by using the fact that a quadratic with roots \( r_1 \) and \( r_2 \) can be expressed as \( (x - r_1)(x - r_2) \). Here, substituting the roots yields \( (x - 1)(x - 5) \). Now, expanding this expression gives: \[ (x - 1)(x - 5) = x^2 - 5x - x + 5 = x^2 - 6x + 5. \] So, the quadratic in standard form is \( x^2 - 6x + 5 \). If you'd like an extra nugget of knowledge, here it is: Sometimes, factoring back can have its challenges! A common mistake is forgetting to distribute properly when expanding. Always double-check each term to avoid pesky errors – practice makes perfect! And here's another fun fact: Quadratics are all around us! From parabolic trajectories in sports, where a basketball arcs towards the hoop, to bridge designs relying on quadratic curves for strength and stability, they truly illustrate the beauty of math in the real world!