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

solve the quadratic equation.\n5x² - 2x - 1 = 0\nwrite one exact, simplified solution in each box. you can add or remove boxes. if a solution is not a real number, write it in the form a + bi or a - bi, where a and b are real numbers.

solve the quadratic equation.\n5x² - 2x - 1 = 0\nwrite one exact, simplified solution in each box. you can add or remove boxes. if a solution is not a real number, write it in the form a + bi or a - bi, where a and b are real numbers.

Answer

Explanation:

Step1: Identify coefficients

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

Step2: Apply quadratic formula

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

Step3: Simplify the square - root and fraction

We know that $\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}$. So $x=\frac{2\pm2\sqrt{6}}{10}=\frac{1\pm\sqrt{6}}{5}$.

Answer:

$x_1=\frac{1+\sqrt{6}}{5}$, $x_2=\frac{1 - \sqrt{6}}{5}$