solve the quadratic equation.\n5x² - 2x - 1 = 0\nwrite one exact solution in each box. you can add or remove…

Explanation:

Step1: Identify coefficients

For the quadratic equation $5x^{2}-2x - 1=0$, we have $a = 5$, $b=-2$, $c=-1$.

Step2: Use quadratic formula

The quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute $a = 5$, $b=-2$, $c=-1$ into it: [ \begin{align*} x&=\frac{-(-2)\pm\sqrt{(-2)^{2}-4\times5\times(-1)}}{2\times5}\ &=\frac{2\pm\sqrt{4 + 20}}{10}\ &=\frac{2\pm\sqrt{24}}{10}\ &=\frac{2\pm2\sqrt{6}}{10}\ &=\frac{1\pm\sqrt{6}}{5} \end{align*} ]

Answer:

$\frac{1+\sqrt{6}}{5}$, $\frac{1 - \sqrt{6}}{5}$