Question

solve the quadratic equation. $$x^{2}-4 x-15=0$$

Short answer

The solutions to the equation \(x^{2}-4x-15=0\) are \(x = 2 + \sqrt{19}\) and \(x = 2 - \sqrt{19}\)

Step-by-step solution

Identify the coefficients a, b, and c

From the quadratic equation \(x^{2}-4x-15=0\), the coefficients \(a\), \(b\), and \(c\) are as follows: \(a = 1\), \(b = -4\), and \(c = -15\).

Substitute the coefficients into the quadratic formula

The quadratic formula is \(\frac{-b ± \sqrt{b^{2}-4ac}}{2a}\). Substituting the values of \(a\), \(b\), and \(c\) we get: \(\frac{4 ± \sqrt{(-4)^{2}-4(1)(-15)}}{2(1)}\)

Simplify the equation

Simplify the equation to find the roots of the quadratic: \(\frac{4 ± \sqrt{16+60}}{2}\), which simplifies to \(\frac{4 ± \sqrt{76}}{2}\) or \(\frac{4 ± 2\sqrt{19}}{2}\)

Solve the equation

Finally, solving the equation gives us the roots: \(x = \frac{4 + 2\sqrt{19}}{2} = 2 + \sqrt{19}\) and \(x = \frac{4 - 2\sqrt{19}}{2} = 2 - \sqrt{19}\)