Question

Use the quadratic formula to solve the equation. $$-2 d^{2}-5 d+19=0$$

Short answer

The roots of the equation \(-2 d^{2}-5 d+19 = 0\) are \(d=\frac{5+\sqrt{177}}{-4}\) and \(d=\frac{5-\sqrt{177}}{-4}\)

Step-by-step solution

Identify the coefficients

From the quadratic equation, identify the coefficients of \(d^{2}\), \(d\) and the constant as \(a = -2\), \(b = -5\) and \(c = 19\), respectively.

Substitute the coefficients into the quadratic formula

Substitute the coefficients \(a\), \(b\), and \(c\) into the quadratic formula \(d=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Thus the formula becomes \(d=\frac{-(-5)\pm\sqrt{(-5)^{2}-4*(-2)*19}}{2*(-2)}\).

Simplify the equation

Simplify the equation further to get the roots. Hence, we get the roots as \(d=\frac{5\pm\sqrt{25+152}}{-4}\). Now the equation simplifies to \(d=\frac{5\pm\sqrt{177}}{-4}\). Therefore, the roots of the quadratic equation are \(d=\frac{5+\sqrt{177}}{-4}\) and \(d=\frac{5-\sqrt{177}}{-4}\).