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Simplifying -0.5 * 32x2 + 320x + 0 = 1000 Multiply -0.5 * 32 -16x2 + 320x + 0 = 1000 Reorder the terms: 0 + 320x + -16x2 = 1000 Remove the zero: 320x + -16x2 = 1000 Solving 320x + -16x2 = 1000 Solving for variable 'x'. Reorder the terms: -1000 + 320x + -16x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 320x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-125 + 40x + -2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-125 + 40x + -2x2)' equal to zero and attempt to solve: Simplifying -125 + 40x + -2x2 = 0 Solving -125 + 40x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 62.5 + -20x + x2 = 0 Move the constant term to the right: Add '-62.5' to each side of the equation. 62.5 + -20x + -62.5 + x2 = 0 + -62.5 Reorder the terms: 62.5 + -62.5 + -20x + x2 = 0 + -62.5 Combine like terms: 62.5 + -62.5 = 0.0 0.0 + -20x + x2 = 0 + -62.5 -20x + x2 = 0 + -62.5 Combine like terms: 0 + -62.5 = -62.5 -20x + x2 = -62.5 The x term is -20x. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20x + 100 + x2 = -62.5 + 100 Reorder the terms: 100 + -20x + x2 = -62.5 + 100 Combine like terms: -62.5 + 100 = 37.5 100 + -20x + x2 = 37.5 Factor a perfect square on the left side: (x + -10)(x + -10) = 37.5 Calculate the square root of the right side: 6.123724357 Break this problem into two subproblems by setting (x + -10) equal to 6.123724357 and -6.123724357.Subproblem 1
x + -10 = 6.123724357 Simplifying x + -10 = 6.123724357 Reorder the terms: -10 + x = 6.123724357 Solving -10 + x = 6.123724357 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x = 6.123724357 + 10 Combine like terms: -10 + 10 = 0 0 + x = 6.123724357 + 10 x = 6.123724357 + 10 Combine like terms: 6.123724357 + 10 = 16.123724357 x = 16.123724357 Simplifying x = 16.123724357Subproblem 2
x + -10 = -6.123724357 Simplifying x + -10 = -6.123724357 Reorder the terms: -10 + x = -6.123724357 Solving -10 + x = -6.123724357 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x = -6.123724357 + 10 Combine like terms: -10 + 10 = 0 0 + x = -6.123724357 + 10 x = -6.123724357 + 10 Combine like terms: -6.123724357 + 10 = 3.876275643 x = 3.876275643 Simplifying x = 3.876275643Solution
The solution to the problem is based on the solutions from the subproblems. x = {16.123724357, 3.876275643}Solution
x = {16.123724357, 3.876275643}
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