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Simplifying (3x + -7) * 2 + -5(2x + 1)(x + -2) = -1x + -1(-1(3x + 1)) Reorder the terms: (-7 + 3x) * 2 + -5(2x + 1)(x + -2) = -1x + -1(-1(3x + 1)) Reorder the terms for easier multiplication: 2(-7 + 3x) + -5(2x + 1)(x + -2) = -1x + -1(-1(3x + 1)) (-7 * 2 + 3x * 2) + -5(2x + 1)(x + -2) = -1x + -1(-1(3x + 1)) (-14 + 6x) + -5(2x + 1)(x + -2) = -1x + -1(-1(3x + 1)) Reorder the terms: -14 + 6x + -5(1 + 2x)(x + -2) = -1x + -1(-1(3x + 1)) Reorder the terms: -14 + 6x + -5(1 + 2x)(-2 + x) = -1x + -1(-1(3x + 1)) Multiply (1 + 2x) * (-2 + x) -14 + 6x + -5(1(-2 + x) + 2x * (-2 + x)) = -1x + -1(-1(3x + 1)) -14 + 6x + -5((-2 * 1 + x * 1) + 2x * (-2 + x)) = -1x + -1(-1(3x + 1)) -14 + 6x + -5((-2 + 1x) + 2x * (-2 + x)) = -1x + -1(-1(3x + 1)) -14 + 6x + -5(-2 + 1x + (-2 * 2x + x * 2x)) = -1x + -1(-1(3x + 1)) -14 + 6x + -5(-2 + 1x + (-4x + 2x2)) = -1x + -1(-1(3x + 1)) Combine like terms: 1x + -4x = -3x -14 + 6x + -5(-2 + -3x + 2x2) = -1x + -1(-1(3x + 1)) -14 + 6x + (-2 * -5 + -3x * -5 + 2x2 * -5) = -1x + -1(-1(3x + 1)) -14 + 6x + (10 + 15x + -10x2) = -1x + -1(-1(3x + 1)) Reorder the terms: -14 + 10 + 6x + 15x + -10x2 = -1x + -1(-1(3x + 1)) Combine like terms: -14 + 10 = -4 -4 + 6x + 15x + -10x2 = -1x + -1(-1(3x + 1)) Combine like terms: 6x + 15x = 21x -4 + 21x + -10x2 = -1x + -1(-1(3x + 1)) Reorder the terms: -4 + 21x + -10x2 = -1x + -1(-1(1 + 3x)) -4 + 21x + -10x2 = -1x + -1((1 * -1 + 3x * -1)) -4 + 21x + -10x2 = -1x + -1((-1 + -3x)) -4 + 21x + -10x2 = -1x + (-1 * -1 + -3x * -1) -4 + 21x + -10x2 = -1x + (1 + 3x) Reorder the terms: -4 + 21x + -10x2 = 1 + -1x + 3x Combine like terms: -1x + 3x = 2x -4 + 21x + -10x2 = 1 + 2x Solving -4 + 21x + -10x2 = 1 + 2x Solving for variable 'x'. Reorder the terms: -4 + -1 + 21x + -2x + -10x2 = 1 + 2x + -1 + -2x Combine like terms: -4 + -1 = -5 -5 + 21x + -2x + -10x2 = 1 + 2x + -1 + -2x Combine like terms: 21x + -2x = 19x -5 + 19x + -10x2 = 1 + 2x + -1 + -2x Reorder the terms: -5 + 19x + -10x2 = 1 + -1 + 2x + -2x Combine like terms: 1 + -1 = 0 -5 + 19x + -10x2 = 0 + 2x + -2x -5 + 19x + -10x2 = 2x + -2x Combine like terms: 2x + -2x = 0 -5 + 19x + -10x2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. 0.5 + -1.9x + x2 = 0.0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.9x + -0.5 + x2 = 0.0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.9x + x2 = 0.0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.9x + x2 = 0.0 + -0.5 -1.9x + x2 = 0.0 + -0.5 Combine like terms: 0.0 + -0.5 = -0.5 -1.9x + x2 = -0.5 The x term is -1.9x. Take half its coefficient (-0.95). Square it (0.9025) and add it to both sides. Add '0.9025' to each side of the equation. -1.9x + 0.9025 + x2 = -0.5 + 0.9025 Reorder the terms: 0.9025 + -1.9x + x2 = -0.5 + 0.9025 Combine like terms: -0.5 + 0.9025 = 0.4025 0.9025 + -1.9x + x2 = 0.4025 Factor a perfect square on the left side: (x + -0.95)(x + -0.95) = 0.4025 Calculate the square root of the right side: 0.634428877 Break this problem into two subproblems by setting (x + -0.95) equal to 0.634428877 and -0.634428877.Subproblem 1
x + -0.95 = 0.634428877 Simplifying x + -0.95 = 0.634428877 Reorder the terms: -0.95 + x = 0.634428877 Solving -0.95 + x = 0.634428877 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.95' to each side of the equation. -0.95 + 0.95 + x = 0.634428877 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + x = 0.634428877 + 0.95 x = 0.634428877 + 0.95 Combine like terms: 0.634428877 + 0.95 = 1.584428877 x = 1.584428877 Simplifying x = 1.584428877Subproblem 2
x + -0.95 = -0.634428877 Simplifying x + -0.95 = -0.634428877 Reorder the terms: -0.95 + x = -0.634428877 Solving -0.95 + x = -0.634428877 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.95' to each side of the equation. -0.95 + 0.95 + x = -0.634428877 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + x = -0.634428877 + 0.95 x = -0.634428877 + 0.95 Combine like terms: -0.634428877 + 0.95 = 0.315571123 x = 0.315571123 Simplifying x = 0.315571123Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.584428877, 0.315571123}
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