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Simplifying 5 = -16x2 + 22x Reorder the terms: 5 = 22x + -16x2 Solving 5 = 22x + -16x2 Solving for variable 'x'. Reorder the terms: 5 + -22x + 16x2 = 22x + -22x + -16x2 + 16x2 Combine like terms: 22x + -22x = 0 5 + -22x + 16x2 = 0 + -16x2 + 16x2 5 + -22x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 5 + -22x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.3125 + -1.375x + x2 = 0 Move the constant term to the right: Add '-0.3125' to each side of the equation. 0.3125 + -1.375x + -0.3125 + x2 = 0 + -0.3125 Reorder the terms: 0.3125 + -0.3125 + -1.375x + x2 = 0 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + -1.375x + x2 = 0 + -0.3125 -1.375x + x2 = 0 + -0.3125 Combine like terms: 0 + -0.3125 = -0.3125 -1.375x + x2 = -0.3125 The x term is -1.375x. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375x + 0.47265625 + x2 = -0.3125 + 0.47265625 Reorder the terms: 0.47265625 + -1.375x + x2 = -0.3125 + 0.47265625 Combine like terms: -0.3125 + 0.47265625 = 0.16015625 0.47265625 + -1.375x + x2 = 0.16015625 Factor a perfect square on the left side: (x + -0.6875)(x + -0.6875) = 0.16015625 Calculate the square root of the right side: 0.400195265 Break this problem into two subproblems by setting (x + -0.6875) equal to 0.400195265 and -0.400195265.Subproblem 1
x + -0.6875 = 0.400195265 Simplifying x + -0.6875 = 0.400195265 Reorder the terms: -0.6875 + x = 0.400195265 Solving -0.6875 + x = 0.400195265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + x = 0.400195265 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + x = 0.400195265 + 0.6875 x = 0.400195265 + 0.6875 Combine like terms: 0.400195265 + 0.6875 = 1.087695265 x = 1.087695265 Simplifying x = 1.087695265Subproblem 2
x + -0.6875 = -0.400195265 Simplifying x + -0.6875 = -0.400195265 Reorder the terms: -0.6875 + x = -0.400195265 Solving -0.6875 + x = -0.400195265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + x = -0.400195265 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + x = -0.400195265 + 0.6875 x = -0.400195265 + 0.6875 Combine like terms: -0.400195265 + 0.6875 = 0.287304735 x = 0.287304735 Simplifying x = 0.287304735Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.087695265, 0.287304735}
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