Question

solve the quadratic equation.
5x² - 2x - 1 = 0
write 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.

Explanation:

Step1: Identify coefficients

For \(5x^2 - 2x - 1 = 0\), \(a = 5\), \(b = -2\), \(c = -1\).

Step2: Use quadratic formula

Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-1)}}{2(5)}\)
Simplify: \(x = \frac{2 \pm \sqrt{4 + 20}}{10} = \frac{2 \pm \sqrt{24}}{10}\)
Simplify \(\sqrt{24}\): \(\sqrt{24} = 2\sqrt{6}\)
So, \(x = \frac{2 \pm 2\sqrt{6}}{10} = \frac{1 \pm \sqrt{6}}{5}\)

Answer:

\(\frac{1 + \sqrt{6}}{5}\), \(\frac{1 - \sqrt{6}}{5}\)