solve the equation by completing the square.\n5x(x + 6) = -50\nx = □ and x = □

Explanation:

Step1: Simplify the equation

First, expand the left - hand side: (5x(x + 6)=5x^{2}+30x). The equation becomes (5x^{2}+30x=-50). Divide each term by (5): (x^{2}+6x=-10).

Step2: Complete the square

For the quadratic expression (x^{2}+bx), to complete the square, we add ((\frac{b}{2})^{2}). Here (b = 6), so ((\frac{6}{2})^{2}=9). Add (9) to both sides of the equation (x^{2}+6x=-10). (x^{2}+6x+9=(x + 3)^{2}). Add (9) to both sides of the equation (x^{2}+6x=-10): (x^{2}+6x+9=-10 + 9). ((x + 3)^{2}=-1). Since (x^{2}+6x+9=(x + 3)^{2}), the equation (x^{2}+6x+9=(x + 3)^{2}). The equation (x^{2}+6x+9=-10 + 9) becomes ((x + 3)^{2}=-1).

Step3: Solve for (x)

Take the square root of both sides: (x+3=\pm\sqrt{-1}). We know that (\sqrt{-1}=i). ((x + 3)^{2}=-1). Take the square root of both sides: (x+3=\pm i). Subtract (3) from both sides: (x=-3\pm i).

Answer:

(x=-3 + i) and (x=-3 - i)