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Simplifying x = -16x2 + 40x + 1 Reorder the terms: x = 1 + 40x + -16x2 Solving x = 1 + 40x + -16x2 Solving for variable 'x'. Reorder the terms: -1 + x + -40x + 16x2 = 1 + 40x + -16x2 + -1 + -40x + 16x2 Combine like terms: x + -40x = -39x -1 + -39x + 16x2 = 1 + 40x + -16x2 + -1 + -40x + 16x2 Reorder the terms: -1 + -39x + 16x2 = 1 + -1 + 40x + -40x + -16x2 + 16x2 Combine like terms: 1 + -1 = 0 -1 + -39x + 16x2 = 0 + 40x + -40x + -16x2 + 16x2 -1 + -39x + 16x2 = 40x + -40x + -16x2 + 16x2 Combine like terms: 40x + -40x = 0 -1 + -39x + 16x2 = 0 + -16x2 + 16x2 -1 + -39x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -1 + -39x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.0625 + -2.4375x + x2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -2.4375x + 0.0625 + x2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -2.4375x + x2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -2.4375x + x2 = 0 + 0.0625 -2.4375x + x2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -2.4375x + x2 = 0.0625 The x term is -2.4375x. Take half its coefficient (-1.21875). Square it (1.485351563) and add it to both sides. Add '1.485351563' to each side of the equation. -2.4375x + 1.485351563 + x2 = 0.0625 + 1.485351563 Reorder the terms: 1.485351563 + -2.4375x + x2 = 0.0625 + 1.485351563 Combine like terms: 0.0625 + 1.485351563 = 1.547851563 1.485351563 + -2.4375x + x2 = 1.547851563 Factor a perfect square on the left side: (x + -1.21875)(x + -1.21875) = 1.547851563 Calculate the square root of the right side: 1.244126828 Break this problem into two subproblems by setting (x + -1.21875) equal to 1.244126828 and -1.244126828.Subproblem 1
x + -1.21875 = 1.244126828 Simplifying x + -1.21875 = 1.244126828 Reorder the terms: -1.21875 + x = 1.244126828 Solving -1.21875 + x = 1.244126828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + x = 1.244126828 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + x = 1.244126828 + 1.21875 x = 1.244126828 + 1.21875 Combine like terms: 1.244126828 + 1.21875 = 2.462876828 x = 2.462876828 Simplifying x = 2.462876828Subproblem 2
x + -1.21875 = -1.244126828 Simplifying x + -1.21875 = -1.244126828 Reorder the terms: -1.21875 + x = -1.244126828 Solving -1.21875 + x = -1.244126828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + x = -1.244126828 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + x = -1.244126828 + 1.21875 x = -1.244126828 + 1.21875 Combine like terms: -1.244126828 + 1.21875 = -0.025376828 x = -0.025376828 Simplifying x = -0.025376828Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.462876828, -0.025376828}
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