Question

Solve the equation by completing the square. $$x^{2}+\frac{3}{5} x-1=0$$

Short answer

The solutions for the equation are \(-\frac{3 + \sqrt{109}}{10}\) and \(-\frac{3 - \sqrt{109}}{10}\).

Step-by-step solution

Equation Setup

Given equation is \(x^2 + \frac{3}{5}x - 1 = 0\). Separating the constant term from other terms, the equation is rewritten as \(x^2 + \frac{3}{5}x = 1\).

Completing The Square

To complete the square, the square of half of the coefficient of \(x\) is added and subtracted on both sides of equation. So, \((1/2 * 3/5)^2 = (3/10)^2 = 9/100\). Thus, the equation becomes: \(x^2 + \frac{3}{5}x + \frac{9}{100} = 1 + \frac{9}{100}\), which simplifies to \(x^2 + \frac{3}{5}x + \frac{9}{100} = \frac{109}{100}\)

Solving for x

Now, we can rewrite the left hand side as a square of binomial. The equation is simplified to \((x + \frac{3}{10})^2 = \frac{109}{100}\). Taking square root on both sides, we obtain \(x + \frac{3}{10} = \pm\sqrt{\frac{109}{100}}\). Thus, the solutions will be \(x = -\frac{3}{10} \pm \sqrt{\frac{109}{100}}\)

Simplifying The Result

After simplifying, we get \(x = -\frac{3}{10} \pm \frac{\sqrt{109}}{10}\). Therefore, the roots of the equation are \(x1 = -\frac{3 + \sqrt{109}}{10}\) and \(x2 = -\frac{3 - \sqrt{109}}{10}\).