solve for the roots in simplest form using the quadratic formula: 4x² + 20x = -17

Explanation:

Step1: Rewrite the equation in standard form

$$4x^{2}+20x + 17=0$$ For a quadratic equation (ax^{2}+bx + c = 0), here (a = 4), (b=20), (c = 17).

Step2: Apply the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a})

First, calculate the discriminant (\Delta=b^{2}-4ac=(20)^{2}-4\times4\times17). [ \begin{align*} \Delta&=400-272\ &=128 \end{align*} ] Then, (x=\frac{-20\pm\sqrt{128}}{2\times4}). Since (\sqrt{128}=\sqrt{64\times2}=8\sqrt{2}), we have (x=\frac{-20\pm8\sqrt{2}}{8}). Simplify the fraction: (x=\frac{-5\pm2\sqrt{2}}{2})

Answer:

(x=\frac{-5 + 2\sqrt{2}}{2}) and (x=\frac{-5 - 2\sqrt{2}}{2})