Question

Solve the equation by completing the square. $$20 x^{2}-120 x-109=0$$

Short answer

The solutions to the equation \(20 x^{2}-120 x-109=0\) by completing the square are \(x = 3 \pm \sqrt{\frac{139}{20}}\)

Step-by-step solution

Simplify the equation

Divide the entire equation by 20 to simplify it. We are then left with \(x^{2}-6x-\frac{109}{20}=0\)

Rearrange the equation

Rearrange the equation in preparation for completing the square. Put the constant term on the right side of the equation which gives us \(x^{2}-6x = \frac{109}{20}\)

Make a perfect square on the left side

Now, make a perfect square on the left side of the equation. Do this by taking half of the coefficient of x, squaring it and adding it to both sides. This gives us \((x-3)^{2}=\frac{109}{20}+3^{2}\) which simplifies to \((x-3)^{2}=\frac{139}{20}\)

Solve for x

Take the square root of both sides of the equation to solve for x. Remember that there will be two solutions, one positive and one negative. This results in \(x-3=\pm \sqrt{\frac{139}{20}}\). To find the final values of x, add 3 to both sides of the equation. Therefore the solutions are \(x = 3 \pm \sqrt{\frac{139}{20}}\)