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Simplifying 32x2 + -20x + 176 = 1000 Reorder the terms: 176 + -20x + 32x2 = 1000 Solving 176 + -20x + 32x2 = 1000 Solving for variable 'x'. Reorder the terms: 176 + -1000 + -20x + 32x2 = 1000 + -1000 Combine like terms: 176 + -1000 = -824 -824 + -20x + 32x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -824 + -20x + 32x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-206 + -5x + 8x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-206 + -5x + 8x2)' equal to zero and attempt to solve: Simplifying -206 + -5x + 8x2 = 0 Solving -206 + -5x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -25.75 + -0.625x + x2 = 0 Move the constant term to the right: Add '25.75' to each side of the equation. -25.75 + -0.625x + 25.75 + x2 = 0 + 25.75 Reorder the terms: -25.75 + 25.75 + -0.625x + x2 = 0 + 25.75 Combine like terms: -25.75 + 25.75 = 0.00 0.00 + -0.625x + x2 = 0 + 25.75 -0.625x + x2 = 0 + 25.75 Combine like terms: 0 + 25.75 = 25.75 -0.625x + x2 = 25.75 The x term is -0.625x. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625x + 0.09765625 + x2 = 25.75 + 0.09765625 Reorder the terms: 0.09765625 + -0.625x + x2 = 25.75 + 0.09765625 Combine like terms: 25.75 + 0.09765625 = 25.84765625 0.09765625 + -0.625x + x2 = 25.84765625 Factor a perfect square on the left side: (x + -0.3125)(x + -0.3125) = 25.84765625 Calculate the square root of the right side: 5.084059033 Break this problem into two subproblems by setting (x + -0.3125) equal to 5.084059033 and -5.084059033.Subproblem 1
x + -0.3125 = 5.084059033 Simplifying x + -0.3125 = 5.084059033 Reorder the terms: -0.3125 + x = 5.084059033 Solving -0.3125 + x = 5.084059033 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = 5.084059033 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = 5.084059033 + 0.3125 x = 5.084059033 + 0.3125 Combine like terms: 5.084059033 + 0.3125 = 5.396559033 x = 5.396559033 Simplifying x = 5.396559033Subproblem 2
x + -0.3125 = -5.084059033 Simplifying x + -0.3125 = -5.084059033 Reorder the terms: -0.3125 + x = -5.084059033 Solving -0.3125 + x = -5.084059033 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = -5.084059033 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = -5.084059033 + 0.3125 x = -5.084059033 + 0.3125 Combine like terms: -5.084059033 + 0.3125 = -4.771559033 x = -4.771559033 Simplifying x = -4.771559033Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.396559033, -4.771559033}Solution
x = {5.396559033, -4.771559033}
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