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Simplifying 10 = -16x2 + 19.5x + 6.5 Reorder the terms: 10 = 6.5 + 19.5x + -16x2 Solving 10 = 6.5 + 19.5x + -16x2 Solving for variable 'x'. Combine like terms: 10 + -6.5 = 3.5 3.5 + -19.5x + 16x2 = 6.5 + 19.5x + -16x2 + -6.5 + -19.5x + 16x2 Reorder the terms: 3.5 + -19.5x + 16x2 = 6.5 + -6.5 + 19.5x + -19.5x + -16x2 + 16x2 Combine like terms: 6.5 + -6.5 = 0.0 3.5 + -19.5x + 16x2 = 0.0 + 19.5x + -19.5x + -16x2 + 16x2 3.5 + -19.5x + 16x2 = 19.5x + -19.5x + -16x2 + 16x2 Combine like terms: 19.5x + -19.5x = 0.0 3.5 + -19.5x + 16x2 = 0.0 + -16x2 + 16x2 3.5 + -19.5x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 3.5 + -19.5x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.21875 + -1.21875x + x2 = 0 Move the constant term to the right: Add '-0.21875' to each side of the equation. 0.21875 + -1.21875x + -0.21875 + x2 = 0 + -0.21875 Reorder the terms: 0.21875 + -0.21875 + -1.21875x + x2 = 0 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + -1.21875x + x2 = 0 + -0.21875 -1.21875x + x2 = 0 + -0.21875 Combine like terms: 0 + -0.21875 = -0.21875 -1.21875x + x2 = -0.21875 The x term is -1.21875x. Take half its coefficient (-0.609375). Square it (0.3713378906) and add it to both sides. Add '0.3713378906' to each side of the equation. -1.21875x + 0.3713378906 + x2 = -0.21875 + 0.3713378906 Reorder the terms: 0.3713378906 + -1.21875x + x2 = -0.21875 + 0.3713378906 Combine like terms: -0.21875 + 0.3713378906 = 0.1525878906 0.3713378906 + -1.21875x + x2 = 0.1525878906 Factor a perfect square on the left side: (x + -0.609375)(x + -0.609375) = 0.1525878906 Calculate the square root of the right side: 0.390625 Break this problem into two subproblems by setting (x + -0.609375) equal to 0.390625 and -0.390625.Subproblem 1
x + -0.609375 = 0.390625 Simplifying x + -0.609375 = 0.390625 Reorder the terms: -0.609375 + x = 0.390625 Solving -0.609375 + x = 0.390625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.609375' to each side of the equation. -0.609375 + 0.609375 + x = 0.390625 + 0.609375 Combine like terms: -0.609375 + 0.609375 = 0.000000 0.000000 + x = 0.390625 + 0.609375 x = 0.390625 + 0.609375 Combine like terms: 0.390625 + 0.609375 = 1 x = 1 Simplifying x = 1Subproblem 2
x + -0.609375 = -0.390625 Simplifying x + -0.609375 = -0.390625 Reorder the terms: -0.609375 + x = -0.390625 Solving -0.609375 + x = -0.390625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.609375' to each side of the equation. -0.609375 + 0.609375 + x = -0.390625 + 0.609375 Combine like terms: -0.609375 + 0.609375 = 0.000000 0.000000 + x = -0.390625 + 0.609375 x = -0.390625 + 0.609375 Combine like terms: -0.390625 + 0.609375 = 0.21875 x = 0.21875 Simplifying x = 0.21875Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, 0.21875}
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