Question

Solve the equation by completing the square. \(5 x(x+6)=-50\)

Short answer

The solution to the equation is \(x = -3 + i\) and \(x = -3 - i\).

Step-by-step solution

Simplify the Equation

First, distribute the 5x into the parenthesis. This gives: \(5x^2 + 30x = -50\). Then rearrange the equation to set it to zero: \(5x^2 + 30x + 50 = 0\)

Factor out Constant Terms

To apply the method of completing the square, it's better to work with a coefficient of 1 for \(x^2\). To achieve this, factor out a 5 from all terms to get the equation: \(x^2 + 6x + 10 = 0\)

Completing the Square

To complete the square, find the number that makes \(x^2 + 6x\) a perfect square trinomial. This is done by taking half of the coefficient of x, squaring it and adding it to both sides. So, for this equation \( \frac{6}{2} = 3\) and \(3^2 = 9\). Now, add 9 to both side of the equation to obtain: \(x^2 + 6x + 9 = -1\)

Rewrite Equation as Perfect Square

Rewrite the left side of the equation as \((x+3)^2\). This gives the equation: \((x + 3)^2 = - 1\)

Solve for x

Take the square root of both sides. Don't forget that solving for x gives two solutions; one positive and one negative. This gives the solution: \(x+3 = \pm \sqrt{-1}\). This means, \(x = -3 \pm \sqrt{-1}\). The symbol indicates that there are two solutions: \(x = -3 + \sqrt{-1}\) and \(x = -3 - \sqrt{-1}\). However, the square root of -1 is undefined in the real number system, and is expressed as the imaginary number, i. Therefore, the solutions are \(x = -3 + i\) and \(x = -3 - i\).