Question

Solve the equation by completing the square. $$x^{2}+10 x=39$$

Short answer

The solutions of the equation are \(x = 3\) and \(x = -13\)

Step-by-step solution

Rearrange the equation

Subtract 39 from both sides to rewrite the equation as \(x^2 + 10x - 39 = 0\)

Complete the square

To complete the square, a number should be added and subtracted on both sides that will make the left side of the equation a perfect square. The number added is the square of half the coefficient of \(x\), which in this case, is \((10/2)^2=25\), that makes the equation as \(x^2 + 10x + 25 - 64 = 0\)

Solve for x

Now the left side of the equation can be written as the square of a binomial, so the equation becomes \((x+5)^2 = 64\). Take the square root of both sides, remembering to consider both positive and negative square roots of 64, gives \(x+5 = ± 8\). This leads to the following two solutions for \(x\): \(x = 8 - 5 = 3\) or \(x = -8 - 5 = -13\)