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
Let's solve these quadratic equations using the method of completing the square. (f) \(x^2 + 6x + 8 = 0\) Step 1: Move the constant term to the right side of the equation. x^2 + 6x = -8 Step 2: Take half of the coefficient of \(x\) (which is \(6\)), square it (\((6/2)^2 = 3^2 = 9\)), and add it to both sides. x^2 + 6x + 9 = -8 + 9 Step 3: Factor the left side as a perfect square trinomial and simplify the right side. (x + 3)^2 = 1 Step 4: Take the square root of both sides. x + 3 = ±sqrt(1) x + 3 = ± 1 Step 5: Solve for \(x\). x = -3 ± 1 x_1 = -3 + 1 = -2 x_2 = -3 - 1 = -4 The roots are \(-2, -4\). --- (g) \(x^2 - 4x - 5 = 0\) Step 1: Move the constant term to the right side of the equation. x^2 - 4x = 5 Step 2: Take half of the coefficient of \(x\) (which is \(-4\)), square it (\((-4/2)^2 = (-2)^2 = 4\)), and add it to both sides. x^2 - 4x + 4 = 5 + 4 Step 3: Factor the left side as a perfect square trinomial and simplify the right side. (x - 2)^2 = 9 Step 4: Take the square root of both sides. x - 2 = ±sqrt(9) x - 2 = ± 3 Step 5: Solve for \(x\). x = 2 ± 3 x_1 = 2 + 3 = 5 x_2 = 2 - 3 = -1 The roots are \(5, -1\). --- (h) \(x^2 - 12x + 35 = 0\) Step 1: Move the constant term to the right side of the equation. x^2 - 12x = -35 Step 2: Take half of the coefficient of \(x\) (which is \(-12\)), square it (\((-12/2)^2 = (-6)^2 = 36\)), and add it to both sides. x^2 - 12x + 36 = -35 + 36 Step 3: Factor the left side as a perfect square trinomial and simplify the right side. (x - 6)^2 = 1 Step 4: Take the square root of both sides. x - 6 = ±sqrt(1) x - 6 = ± 1 Step 5: Solve for \(x\). x = 6 ± 1 x_1 = 6 + 1 = 7 x_2 = 6 - 1 = 5 The roots are \(7, 5\). --- (i) \(3x^2 - 12x + 7 = 0\) Step 1: Divide the entire equation by the coefficient of \(x^2\) (which is \(3\)). x^2 - 4x + (7)/(3) = 0 Step 2: Move the constant term to the right side of the equation. x^2 - 4x = -(7)/(3) Step 3: Take half of the coefficient of \(x\) (which is \(-4\)), square it (\((-4/2)^2 = (-2)^2 = 4\)), and add it to both sides. x^2 - 4x + 4 = -(7)/(3) + 4 x^2 - 4x + 4 = -(7)/(3) + (12)/(3) x^2 - 4x + 4 = (5)/(3) Step 4: Factor the left side as a perfect square trinomial. (x - 2)^2 = (5)/(3) Step 5: Take the square root of both sides. x - 2 = ±sqrt((5)/(3)) x - 2 = ±sqrt(5)sqrt(3) x - 2 = ±sqrt(5)sqrt(3)3 x - 2 = ±sqrt(15)3 Step 6: Solve for \(x\). x = 2 ±sqrt(15)3 x = 6 ± sqrt(15)3 The roots are \(6 ± sqrt(15)3\). --- (j) \(2x^2 + 8x + 5 = 0\) Step 1: Divide the entire equation by the coefficient of \(x^2\) (which is \(2\)). x^2 + 4x + (5)/(2) = 0 Step 2: Move the constant term to the right side of the equation. x^2 + 4x = -(5)/(2) Step 3: Take half of the coefficient of \(x\) (which is \(4\)), square it (\((4/2)^2 = 2^2 = 4\)), and add it to both sides. x^2 + 4x + 4 = -(5)/(2) + 4 x^2 + 4x + 4 = -(5)/(2) + (8)/(2) x^2 + 4x + 4 = (3)/(2) Step 4: Factor the left side as a perfect square trinomial. (x + 2)^2 = (3)/(2) Step 5: Take the square root of both sides. x + 2 = ±sqrt((3)/(2)) x + 2 = ±sqrt(3)sqrt(2) x + 2 = ±sqrt(3)sqrt(2)2 x + 2 = ±sqrt(6)2 Step 6: Solve for \(x\). x = -2 ±sqrt(6)2 x = -4 ± sqrt(6)2 The roots are \(-4 ± sqrt(6)2\). 3 done, 2 left today. You're making progress.