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
To solve the equation x^2 + 11x - 24 = 0 using the method of completing the square: Step 1: Move the constant term to the right side of the equation. x^2 + 11x = 24 Step 2: Take half of the coefficient of the x term, square it, and add it to both sides of the equation. The coefficient of x is 11. Half of 11 is (11)/(2). Squaring it gives ((11)/(2))^2 = (121)/(4). Add (121)/(4) to both sides: x^2 + 11x + (121)/(4) = 24 + (121)/(4) Step 3: Factor the left side as a perfect square trinomial and simplify the right side. The left side factors as (x + (11)/(2))^2. For the right side, find a common denominator: 24 = (24 × 4)/(4) = (96)/(4). So, 24 + (121)/(4) = (96)/(4) + (121)/(4) = (96 + 121)/(4) = (217)/(4). The equation becomes: (x + (11)/(2))^2 = (217)/(4) Step 4: Take the square root of both sides. x + (11)/(2) = ±sqrt((217)/(4)) x + (11)/(2) = ±sqrt(217)sqrt(4) x + (11)/(2) = ±sqrt(217)2 Step 5: Solve for x. x = -(11)/(2) ± sqrt(217)2 x = -11 ± sqrt(217)2 The solutions are: x_1 = -11 + sqrt(217)2 x_2 = -11 - sqrt(217)2 The final answers are x = -11 ± sqrt(217)2.