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Simplifying 22 = -16x2 + 40x Reorder the terms: 22 = 40x + -16x2 Solving 22 = 40x + -16x2 Solving for variable 'x'. Reorder the terms: 22 + -40x + 16x2 = 40x + -40x + -16x2 + 16x2 Combine like terms: 40x + -40x = 0 22 + -40x + 16x2 = 0 + -16x2 + 16x2 22 + -40x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 22 + -40x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(11 + -20x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(11 + -20x + 8x2)' equal to zero and attempt to solve: Simplifying 11 + -20x + 8x2 = 0 Solving 11 + -20x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.375 + -2.5x + x2 = 0 Move the constant term to the right: Add '-1.375' to each side of the equation. 1.375 + -2.5x + -1.375 + x2 = 0 + -1.375 Reorder the terms: 1.375 + -1.375 + -2.5x + x2 = 0 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + -2.5x + x2 = 0 + -1.375 -2.5x + x2 = 0 + -1.375 Combine like terms: 0 + -1.375 = -1.375 -2.5x + x2 = -1.375 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = -1.375 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -1.375 + 1.5625 Combine like terms: -1.375 + 1.5625 = 0.1875 1.5625 + -2.5x + x2 = 0.1875 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (x + -1.25) equal to 0.433012702 and -0.433012702.Subproblem 1
x + -1.25 = 0.433012702 Simplifying x + -1.25 = 0.433012702 Reorder the terms: -1.25 + x = 0.433012702 Solving -1.25 + x = 0.433012702 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 0.433012702 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.433012702 + 1.25 x = 0.433012702 + 1.25 Combine like terms: 0.433012702 + 1.25 = 1.683012702 x = 1.683012702 Simplifying x = 1.683012702Subproblem 2
x + -1.25 = -0.433012702 Simplifying x + -1.25 = -0.433012702 Reorder the terms: -1.25 + x = -0.433012702 Solving -1.25 + x = -0.433012702 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -0.433012702 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.433012702 + 1.25 x = -0.433012702 + 1.25 Combine like terms: -0.433012702 + 1.25 = 0.816987298 x = 0.816987298 Simplifying x = 0.816987298Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.683012702, 0.816987298}Solution
x = {1.683012702, 0.816987298}
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