Question

Use the quadratic formula to solve the equation. $$7 x^{2}+2 x-1=0$$

Short answer

The solutions to the equation \(7x^2 + 2x - 1 =0\) are \(x = -\frac{1}{7} + \frac{2\sqrt{2}}{7}\) and \(x = -\frac{1}{7} - \frac{2\sqrt{2}}{7}\).

Step-by-step solution

Identify a, b and c values

Compare the given quadratic equation \(7x^2 + 2x - 1 =0\) with the standard form. For this equation, the coefficients are \(a = 7\), \(b = 2\), and \(c = -1\).

Plug in the values into the quadratic formula

We substitute \(a\), \(b\), and \(c\) into the quadratic formula. This gives us: \(x = \frac{-2 \pm \sqrt{2^2-4*7*(-1)}}{2*7}\)

Simplify the expression under the square root

This step requires us to calculate the discriminant, \(b^2-4ac\). We obtain: \(x = \frac{-2 \pm \sqrt{4+28}}{14}\)

Simplify the entire equation

We find: \(x = \frac{-2 \pm \sqrt{32}}{14} = \frac{-2 \pm 4\sqrt{2}}{14}\).

Simplify the solutions

Finally, simplify to find: \(x = -\frac{1}{7} \pm \frac{2\sqrt{2}}{7}\)