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
Here are the solutions to the quadratic equations. 1) (y + (1)/(4))^2 = (9)/(16) Step 1: Take the square root of both sides. y + (1)/(4) = ±sqrt((9)/(16)) y + (1)/(4) = ±(3)/(4) Step 2: Solve for y. y = -(1)/(4) ± (3)/(4) This gives two solutions: y_1 = -(1)/(4) + (3)/(4) = (2)/(4) = (1)/(2) y_2 = -(1)/(4) - (3)/(4) = -(4)/(4) = -1 The solutions are y = (1)/(2), -1. 2) (x + 3)^2 = 49 Step 1: Take the square root of both sides. x + 3 = ±sqrt(49) x + 3 = ± 7 Step 2: Solve for x. x = -3 ± 7 This gives two solutions: x_1 = -3 + 7 = 4 x_2 = -3 - 7 = -10 The solutions are x = 4, -10. 3) 3x^2 - 5x - 2 = 0 Step 1: Factor the quadratic expression. We look for two numbers that multiply to 3 × -2 = -6 and add to -5. These numbers are -6 and 1. 3x^2 - 6x + x - 2 = 0 3x(x - 2) + 1(x - 2) = 0 (3x + 1)(x - 2) = 0 Step 2: Set each factor to zero and solve for x. 3x + 1 = 0 3x = -1 x_1 = -(1)/(3) x - 2 = 0 x_2 = 2 The solutions are x = -(1)/(3), 2. 4) (2x - 1)^2 = 1 Step 1: Take the square root of both sides. 2x - 1 = ±sqrt(1) 2x - 1 = ± 1 Step 2: Solve for x. 2x = 1 ± 1 This gives two solutions: 2x_1 = 1 + 1 = 2 x_1 = 1 2x_2 = 1 - 1 = 0 x_2 = 0 The solutions are x = 1, 0. 5) x^2 + 5x - 24 = 0 Step 1: Factor the quadratic expression. We look for two numbers that multiply to -24 and add to 5. These numbers are 8 and -3. (x + 8)(x - 3) = 0 Step 2: Set each factor to zero and solve for x. x + 8 = 0 x_1 = -8 x - 3 = 0 x_2 = 3 The solutions are x = -8, 3. For x^2 = 16 (from the top right of the board): Step 1: Take the square root of both sides. x = ±sqrt(16) x = ± 4 The solutions are x = 4, -4. 3 done, 2 left today. You're making progress.