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Simplifying -0.5 * 32x2 + 1000x + 50 = 2000 Multiply -0.5 * 32 -16x2 + 1000x + 50 = 2000 Reorder the terms: 50 + 1000x + -16x2 = 2000 Solving 50 + 1000x + -16x2 = 2000 Solving for variable 'x'. Reorder the terms: 50 + -2000 + 1000x + -16x2 = 2000 + -2000 Combine like terms: 50 + -2000 = -1950 -1950 + 1000x + -16x2 = 2000 + -2000 Combine like terms: 2000 + -2000 = 0 -1950 + 1000x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-975 + 500x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-975 + 500x + -8x2)' equal to zero and attempt to solve: Simplifying -975 + 500x + -8x2 = 0 Solving -975 + 500x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 121.875 + -62.5x + x2 = 0 Move the constant term to the right: Add '-121.875' to each side of the equation. 121.875 + -62.5x + -121.875 + x2 = 0 + -121.875 Reorder the terms: 121.875 + -121.875 + -62.5x + x2 = 0 + -121.875 Combine like terms: 121.875 + -121.875 = 0.000 0.000 + -62.5x + x2 = 0 + -121.875 -62.5x + x2 = 0 + -121.875 Combine like terms: 0 + -121.875 = -121.875 -62.5x + x2 = -121.875 The x term is -62.5x. Take half its coefficient (-31.25). Square it (976.5625) and add it to both sides. Add '976.5625' to each side of the equation. -62.5x + 976.5625 + x2 = -121.875 + 976.5625 Reorder the terms: 976.5625 + -62.5x + x2 = -121.875 + 976.5625 Combine like terms: -121.875 + 976.5625 = 854.6875 976.5625 + -62.5x + x2 = 854.6875 Factor a perfect square on the left side: (x + -31.25)(x + -31.25) = 854.6875 Calculate the square root of the right side: 29.235038909 Break this problem into two subproblems by setting (x + -31.25) equal to 29.235038909 and -29.235038909.Subproblem 1
x + -31.25 = 29.235038909 Simplifying x + -31.25 = 29.235038909 Reorder the terms: -31.25 + x = 29.235038909 Solving -31.25 + x = 29.235038909 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '31.25' to each side of the equation. -31.25 + 31.25 + x = 29.235038909 + 31.25 Combine like terms: -31.25 + 31.25 = 0.00 0.00 + x = 29.235038909 + 31.25 x = 29.235038909 + 31.25 Combine like terms: 29.235038909 + 31.25 = 60.485038909 x = 60.485038909 Simplifying x = 60.485038909Subproblem 2
x + -31.25 = -29.235038909 Simplifying x + -31.25 = -29.235038909 Reorder the terms: -31.25 + x = -29.235038909 Solving -31.25 + x = -29.235038909 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '31.25' to each side of the equation. -31.25 + 31.25 + x = -29.235038909 + 31.25 Combine like terms: -31.25 + 31.25 = 0.00 0.00 + x = -29.235038909 + 31.25 x = -29.235038909 + 31.25 Combine like terms: -29.235038909 + 31.25 = 2.014961091 x = 2.014961091 Simplifying x = 2.014961091Solution
The solution to the problem is based on the solutions from the subproblems. x = {60.485038909, 2.014961091}Solution
x = {60.485038909, 2.014961091}
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