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Simplifying (10x + -3)(8 + -5x) = -74 Reorder the terms: (-3 + 10x)(8 + -5x) = -74 Multiply (-3 + 10x) * (8 + -5x) (-3(8 + -5x) + 10x * (8 + -5x)) = -74 ((8 * -3 + -5x * -3) + 10x * (8 + -5x)) = -74 ((-24 + 15x) + 10x * (8 + -5x)) = -74 (-24 + 15x + (8 * 10x + -5x * 10x)) = -74 (-24 + 15x + (80x + -50x2)) = -74 Combine like terms: 15x + 80x = 95x (-24 + 95x + -50x2) = -74 Solving -24 + 95x + -50x2 = -74 Solving for variable 'x'. Reorder the terms: -24 + 74 + 95x + -50x2 = -74 + 74 Combine like terms: -24 + 74 = 50 50 + 95x + -50x2 = -74 + 74 Combine like terms: -74 + 74 = 0 50 + 95x + -50x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(10 + 19x + -10x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(10 + 19x + -10x2)' equal to zero and attempt to solve: Simplifying 10 + 19x + -10x2 = 0 Solving 10 + 19x + -10x2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -1 + -1.9x + x2 = 0.0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1.9x + 1 + x2 = 0.0 + 1 Reorder the terms: -1 + 1 + -1.9x + x2 = 0.0 + 1 Combine like terms: -1 + 1 = 0 0 + -1.9x + x2 = 0.0 + 1 -1.9x + x2 = 0.0 + 1 Combine like terms: 0.0 + 1 = 1 -1.9x + x2 = 1 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 = 1 + 0.9025 Reorder the terms: 0.9025 + -1.9x + x2 = 1 + 0.9025 Combine like terms: 1 + 0.9025 = 1.9025 0.9025 + -1.9x + x2 = 1.9025 Factor a perfect square on the left side: (x + -0.95)(x + -0.95) = 1.9025 Calculate the square root of the right side: 1.379311422 Break this problem into two subproblems by setting (x + -0.95) equal to 1.379311422 and -1.379311422.Subproblem 1
x + -0.95 = 1.379311422 Simplifying x + -0.95 = 1.379311422 Reorder the terms: -0.95 + x = 1.379311422 Solving -0.95 + x = 1.379311422 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 = 1.379311422 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + x = 1.379311422 + 0.95 x = 1.379311422 + 0.95 Combine like terms: 1.379311422 + 0.95 = 2.329311422 x = 2.329311422 Simplifying x = 2.329311422Subproblem 2
x + -0.95 = -1.379311422 Simplifying x + -0.95 = -1.379311422 Reorder the terms: -0.95 + x = -1.379311422 Solving -0.95 + x = -1.379311422 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 = -1.379311422 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + x = -1.379311422 + 0.95 x = -1.379311422 + 0.95 Combine like terms: -1.379311422 + 0.95 = -0.429311422 x = -0.429311422 Simplifying x = -0.429311422Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.329311422, -0.429311422}Solution
x = {2.329311422, -0.429311422}
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