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Simplifying 24 = -16x2 + 50x Reorder the terms: 24 = 50x + -16x2 Solving 24 = 50x + -16x2 Solving for variable 'x'. Reorder the terms: 24 + -50x + 16x2 = 50x + -50x + -16x2 + 16x2 Combine like terms: 50x + -50x = 0 24 + -50x + 16x2 = 0 + -16x2 + 16x2 24 + -50x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 24 + -50x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(12 + -25x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + -25x + 8x2)' equal to zero and attempt to solve: Simplifying 12 + -25x + 8x2 = 0 Solving 12 + -25x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.5 + -3.125x + x2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3.125x + -1.5 + x2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3.125x + x2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3.125x + x2 = 0 + -1.5 -3.125x + x2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3.125x + x2 = -1.5 The x term is -3.125x. Take half its coefficient (-1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. -3.125x + 2.44140625 + x2 = -1.5 + 2.44140625 Reorder the terms: 2.44140625 + -3.125x + x2 = -1.5 + 2.44140625 Combine like terms: -1.5 + 2.44140625 = 0.94140625 2.44140625 + -3.125x + x2 = 0.94140625 Factor a perfect square on the left side: (x + -1.5625)(x + -1.5625) = 0.94140625 Calculate the square root of the right side: 0.970260919 Break this problem into two subproblems by setting (x + -1.5625) equal to 0.970260919 and -0.970260919.Subproblem 1
x + -1.5625 = 0.970260919 Simplifying x + -1.5625 = 0.970260919 Reorder the terms: -1.5625 + x = 0.970260919 Solving -1.5625 + x = 0.970260919 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + x = 0.970260919 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + x = 0.970260919 + 1.5625 x = 0.970260919 + 1.5625 Combine like terms: 0.970260919 + 1.5625 = 2.532760919 x = 2.532760919 Simplifying x = 2.532760919Subproblem 2
x + -1.5625 = -0.970260919 Simplifying x + -1.5625 = -0.970260919 Reorder the terms: -1.5625 + x = -0.970260919 Solving -1.5625 + x = -0.970260919 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + x = -0.970260919 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + x = -0.970260919 + 1.5625 x = -0.970260919 + 1.5625 Combine like terms: -0.970260919 + 1.5625 = 0.592239081 x = 0.592239081 Simplifying x = 0.592239081Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.532760919, 0.592239081}Solution
x = {2.532760919, 0.592239081}
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