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Simplifying 40 = -16x2 + 192x Reorder the terms: 40 = 192x + -16x2 Solving 40 = 192x + -16x2 Solving for variable 'x'. Reorder the terms: 40 + -192x + 16x2 = 192x + -192x + -16x2 + 16x2 Combine like terms: 192x + -192x = 0 40 + -192x + 16x2 = 0 + -16x2 + 16x2 40 + -192x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 40 + -192x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(5 + -24x + 2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(5 + -24x + 2x2)' equal to zero and attempt to solve: Simplifying 5 + -24x + 2x2 = 0 Solving 5 + -24x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + -12x + x2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -12x + -2.5 + x2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -12x + x2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -12x + x2 = 0 + -2.5 -12x + x2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -12x + x2 = -2.5 The x term is -12x. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12x + 36 + x2 = -2.5 + 36 Reorder the terms: 36 + -12x + x2 = -2.5 + 36 Combine like terms: -2.5 + 36 = 33.5 36 + -12x + x2 = 33.5 Factor a perfect square on the left side: (x + -6)(x + -6) = 33.5 Calculate the square root of the right side: 5.787918451 Break this problem into two subproblems by setting (x + -6) equal to 5.787918451 and -5.787918451.Subproblem 1
x + -6 = 5.787918451 Simplifying x + -6 = 5.787918451 Reorder the terms: -6 + x = 5.787918451 Solving -6 + x = 5.787918451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x = 5.787918451 + 6 Combine like terms: -6 + 6 = 0 0 + x = 5.787918451 + 6 x = 5.787918451 + 6 Combine like terms: 5.787918451 + 6 = 11.787918451 x = 11.787918451 Simplifying x = 11.787918451Subproblem 2
x + -6 = -5.787918451 Simplifying x + -6 = -5.787918451 Reorder the terms: -6 + x = -5.787918451 Solving -6 + x = -5.787918451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x = -5.787918451 + 6 Combine like terms: -6 + 6 = 0 0 + x = -5.787918451 + 6 x = -5.787918451 + 6 Combine like terms: -5.787918451 + 6 = 0.212081549 x = 0.212081549 Simplifying x = 0.212081549Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.787918451, 0.212081549}Solution
x = {11.787918451, 0.212081549}
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