This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Move the constant term to the right side of the equation. z^2 + 12z + 52 = 0 z^2 + 12z = -52 Step 2: Complete the square on the left side. To complete the square, take half of the coefficient of z (which is 12), square it, and add it to both sides of the equation. Half of 12 is 6. 6^2 = 36. z^2 + 12z + 36 = -52 + 36 Step 3: Factor the perfect square trinomial on the left side. (z + 6)^2 = -16 Step 4: Take the square root of both sides. sqrt((z + 6)^2) = sqrt(-16) z + 6 = ± sqrt(16) · sqrt(-1) z + 6 = ± 4i Step 5: Solve for z. z = -6 ± 4i The solutions are z = -6 + 4i and z = -6 - 4i. The final answer is z = -6 ± 4i.