\[ 5 x^{2}+x-\frac{1}{2}=0 \] Solve the following equation by completing the square. OA. The solution set is \( \left\{\begin{array}{l}\text { Sorrect choice below and, if necessary, fll in the answer box to complete your choice. } \\ \text { (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. } \\ \text { Use a comma to separate answers as needed.) }\end{array}\right. \) B. There is no solution.

To solve the equation \( 5x^{2} + x - \frac{1}{2} = 0 \) by completing the square, we first rearrange it into the standard form of a quadratic equation: \[ 5x^2 + x = \frac{1}{2} \] Next, we'll our coefficient of \(x^2\) to 1 by dividing all terms by 5: \[ x^2 + \frac{1}{5}x = \frac{1}{10} \] Now, to complete the square, we take half of the coefficient of \(x\) (which is \(\frac{1}{5}\)), square it (\(\left(\frac{1}{10}\right)^2 = \frac{1}{100}\)), and add this to both sides: \[ x^2 + \frac{1}{5}x + \frac{1}{100} = \frac{1}{10} + \frac{1}{100} \] This simplifies to: \[ \left(x + \frac{1}{10}\right)^2 = \frac{10}{100} + \frac{1}{100} = \frac{11}{100} \] Taking the square root of both sides gives us: \[ x + \frac{1}{10} = \pm \frac{\sqrt{11}}{10} \] Therefore, we can isolate \(x\): \[ x = -\frac{1}{10} \pm \frac{\sqrt{11}}{10} \] This can be expressed as the solution set: \[ x = \frac{-1 + \sqrt{11}}{10}, \frac{-1 - \sqrt{11}}{10} \] So, the correct choice is: \[ \left\{ \frac{-1 + \sqrt{11}}{10}, \frac{-1 - \sqrt{11}}{10} \right\} \] --- The concept of completing the square dates back to ancient civilizations, like the Babylonians. They used geometric techniques to solve quadratic equations long before modern algebra was developed! It’s fascinating how through the ages, math has evolved while still holding onto these timeless tricks. Completing the square not only helps in solving quadratics but also provides insights into the properties of parabolas, like identifying the vertex. This method is crucial in calculus, particularly in finding the maximum and minimum values of quadratic functions. So, mastering this technique is a stepping stone to understanding higher math concepts!