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Simplifying 20 = -16x2 + 40x + 6 Reorder the terms: 20 = 6 + 40x + -16x2 Solving 20 = 6 + 40x + -16x2 Solving for variable 'x'. Combine like terms: 20 + -6 = 14 14 + -40x + 16x2 = 6 + 40x + -16x2 + -6 + -40x + 16x2 Reorder the terms: 14 + -40x + 16x2 = 6 + -6 + 40x + -40x + -16x2 + 16x2 Combine like terms: 6 + -6 = 0 14 + -40x + 16x2 = 0 + 40x + -40x + -16x2 + 16x2 14 + -40x + 16x2 = 40x + -40x + -16x2 + 16x2 Combine like terms: 40x + -40x = 0 14 + -40x + 16x2 = 0 + -16x2 + 16x2 14 + -40x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 14 + -40x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -20x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -20x + 8x2)' equal to zero and attempt to solve: Simplifying 7 + -20x + 8x2 = 0 Solving 7 + -20x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.875 + -2.5x + x2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -2.5x + -0.875 + x2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -2.5x + x2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -2.5x + x2 = 0 + -0.875 -2.5x + x2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -2.5x + x2 = -0.875 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 = -0.875 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -0.875 + 1.5625 Combine like terms: -0.875 + 1.5625 = 0.6875 1.5625 + -2.5x + x2 = 0.6875 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 0.6875 Calculate the square root of the right side: 0.829156198 Break this problem into two subproblems by setting (x + -1.25) equal to 0.829156198 and -0.829156198.Subproblem 1
x + -1.25 = 0.829156198 Simplifying x + -1.25 = 0.829156198 Reorder the terms: -1.25 + x = 0.829156198 Solving -1.25 + x = 0.829156198 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 = 0.829156198 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.829156198 + 1.25 x = 0.829156198 + 1.25 Combine like terms: 0.829156198 + 1.25 = 2.079156198 x = 2.079156198 Simplifying x = 2.079156198Subproblem 2
x + -1.25 = -0.829156198 Simplifying x + -1.25 = -0.829156198 Reorder the terms: -1.25 + x = -0.829156198 Solving -1.25 + x = -0.829156198 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 = -0.829156198 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.829156198 + 1.25 x = -0.829156198 + 1.25 Combine like terms: -0.829156198 + 1.25 = 0.420843802 x = 0.420843802 Simplifying x = 0.420843802Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.079156198, 0.420843802}Solution
x = {2.079156198, 0.420843802}
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