Question

Use the quadratic formula to solve the equation. $$7 y^{2}-9 y-17=0$$

Short answer

The solutions to the equation are \(y=\frac{9+ \sqrt{557}}{14}\) and \(y=\frac{9- \sqrt{557}}{14}\).

Step-by-step solution

Identify the coefficients a, b, and c

From the quadratic equation \(7 y^{2}-9 y-17=0\), we can see that \(a=7, b=-9\), and \(c=-17\).

Plug the coefficients into the quadratic formula

Substitute the coefficients a, b, and c into the quadratic formula \(-\frac{b \pm \sqrt{{b^2-4ac}}}{2a}\).\nThis leads to \(-\frac{-9 \pm \sqrt{(-9)^2-4*7*(-17)}}{2*7}\), which simplifies to \(-\frac{-9 \pm \sqrt{81+476}}{14}\), then to \(-\frac{-9 \pm \sqrt{557}}{14}\).

Calculate the roots of the equation

We now have to calculate the roots of the quadratic equation by splitting the formula into the addition and subtraction of the square root. This will lead to two solutions for y: \(-\frac{-9 + \sqrt{557}}{14}\) and \(-\frac{-9 - \sqrt{557}}{14}\), which simplifies to \(y=\frac{9+ \sqrt{557}}{14}\) and \(y=\frac{9- \sqrt{557}}{14}\).