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Simplifying 4 + 2(3x + 1) + -5(2x + -6)(2x + 6) = 8(3x + 6) Reorder the terms: 4 + 2(1 + 3x) + -5(2x + -6)(2x + 6) = 8(3x + 6) 4 + (1 * 2 + 3x * 2) + -5(2x + -6)(2x + 6) = 8(3x + 6) 4 + (2 + 6x) + -5(2x + -6)(2x + 6) = 8(3x + 6) Reorder the terms: 4 + 2 + 6x + -5(-6 + 2x)(2x + 6) = 8(3x + 6) Reorder the terms: 4 + 2 + 6x + -5(-6 + 2x)(6 + 2x) = 8(3x + 6) Multiply (-6 + 2x) * (6 + 2x) 4 + 2 + 6x + -5(-6(6 + 2x) + 2x * (6 + 2x)) = 8(3x + 6) 4 + 2 + 6x + -5((6 * -6 + 2x * -6) + 2x * (6 + 2x)) = 8(3x + 6) 4 + 2 + 6x + -5((-36 + -12x) + 2x * (6 + 2x)) = 8(3x + 6) 4 + 2 + 6x + -5(-36 + -12x + (6 * 2x + 2x * 2x)) = 8(3x + 6) 4 + 2 + 6x + -5(-36 + -12x + (12x + 4x2)) = 8(3x + 6) Combine like terms: -12x + 12x = 0 4 + 2 + 6x + -5(-36 + 0 + 4x2) = 8(3x + 6) 4 + 2 + 6x + -5(-36 + 4x2) = 8(3x + 6) 4 + 2 + 6x + (-36 * -5 + 4x2 * -5) = 8(3x + 6) 4 + 2 + 6x + (180 + -20x2) = 8(3x + 6) Reorder the terms: 4 + 2 + 180 + 6x + -20x2 = 8(3x + 6) Combine like terms: 4 + 2 = 6 6 + 180 + 6x + -20x2 = 8(3x + 6) Combine like terms: 6 + 180 = 186 186 + 6x + -20x2 = 8(3x + 6) Reorder the terms: 186 + 6x + -20x2 = 8(6 + 3x) 186 + 6x + -20x2 = (6 * 8 + 3x * 8) 186 + 6x + -20x2 = (48 + 24x) Solving 186 + 6x + -20x2 = 48 + 24x Solving for variable 'x'. Reorder the terms: 186 + -48 + 6x + -24x + -20x2 = 48 + 24x + -48 + -24x Combine like terms: 186 + -48 = 138 138 + 6x + -24x + -20x2 = 48 + 24x + -48 + -24x Combine like terms: 6x + -24x = -18x 138 + -18x + -20x2 = 48 + 24x + -48 + -24x Reorder the terms: 138 + -18x + -20x2 = 48 + -48 + 24x + -24x Combine like terms: 48 + -48 = 0 138 + -18x + -20x2 = 0 + 24x + -24x 138 + -18x + -20x2 = 24x + -24x Combine like terms: 24x + -24x = 0 138 + -18x + -20x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(69 + -9x + -10x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(69 + -9x + -10x2)' equal to zero and attempt to solve: Simplifying 69 + -9x + -10x2 = 0 Solving 69 + -9x + -10x2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -6.9 + 0.9x + x2 = 0.0 Move the constant term to the right: Add '6.9' to each side of the equation. -6.9 + 0.9x + 6.9 + x2 = 0.0 + 6.9 Reorder the terms: -6.9 + 6.9 + 0.9x + x2 = 0.0 + 6.9 Combine like terms: -6.9 + 6.9 = 0.0 0.0 + 0.9x + x2 = 0.0 + 6.9 0.9x + x2 = 0.0 + 6.9 Combine like terms: 0.0 + 6.9 = 6.9 0.9x + x2 = 6.9 The x term is 0.9x. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9x + 0.2025 + x2 = 6.9 + 0.2025 Reorder the terms: 0.2025 + 0.9x + x2 = 6.9 + 0.2025 Combine like terms: 6.9 + 0.2025 = 7.1025 0.2025 + 0.9x + x2 = 7.1025 Factor a perfect square on the left side: (x + 0.45)(x + 0.45) = 7.1025 Calculate the square root of the right side: 2.665051594 Break this problem into two subproblems by setting (x + 0.45) equal to 2.665051594 and -2.665051594.Subproblem 1
x + 0.45 = 2.665051594 Simplifying x + 0.45 = 2.665051594 Reorder the terms: 0.45 + x = 2.665051594 Solving 0.45 + x = 2.665051594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + x = 2.665051594 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + x = 2.665051594 + -0.45 x = 2.665051594 + -0.45 Combine like terms: 2.665051594 + -0.45 = 2.215051594 x = 2.215051594 Simplifying x = 2.215051594Subproblem 2
x + 0.45 = -2.665051594 Simplifying x + 0.45 = -2.665051594 Reorder the terms: 0.45 + x = -2.665051594 Solving 0.45 + x = -2.665051594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + x = -2.665051594 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + x = -2.665051594 + -0.45 x = -2.665051594 + -0.45 Combine like terms: -2.665051594 + -0.45 = -3.115051594 x = -3.115051594 Simplifying x = -3.115051594Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.215051594, -3.115051594}Solution
x = {2.215051594, -3.115051594}
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