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Simplifying -16x2 + 27x + 4 = 14 Reorder the terms: 4 + 27x + -16x2 = 14 Solving 4 + 27x + -16x2 = 14 Solving for variable 'x'. Reorder the terms: 4 + -14 + 27x + -16x2 = 14 + -14 Combine like terms: 4 + -14 = -10 -10 + 27x + -16x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -10 + 27x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.625 + -1.6875x + x2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + -1.6875x + -0.625 + x2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + -1.6875x + x2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + -1.6875x + x2 = 0 + -0.625 -1.6875x + x2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 -1.6875x + x2 = -0.625 The x term is -1.6875x. Take half its coefficient (-0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. -1.6875x + 0.7119140625 + x2 = -0.625 + 0.7119140625 Reorder the terms: 0.7119140625 + -1.6875x + x2 = -0.625 + 0.7119140625 Combine like terms: -0.625 + 0.7119140625 = 0.0869140625 0.7119140625 + -1.6875x + x2 = 0.0869140625 Factor a perfect square on the left side: (x + -0.84375)(x + -0.84375) = 0.0869140625 Calculate the square root of the right side: 0.29481191 Break this problem into two subproblems by setting (x + -0.84375) equal to 0.29481191 and -0.29481191.Subproblem 1
x + -0.84375 = 0.29481191 Simplifying x + -0.84375 = 0.29481191 Reorder the terms: -0.84375 + x = 0.29481191 Solving -0.84375 + x = 0.29481191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = 0.29481191 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = 0.29481191 + 0.84375 x = 0.29481191 + 0.84375 Combine like terms: 0.29481191 + 0.84375 = 1.13856191 x = 1.13856191 Simplifying x = 1.13856191Subproblem 2
x + -0.84375 = -0.29481191 Simplifying x + -0.84375 = -0.29481191 Reorder the terms: -0.84375 + x = -0.29481191 Solving -0.84375 + x = -0.29481191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = -0.29481191 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = -0.29481191 + 0.84375 x = -0.29481191 + 0.84375 Combine like terms: -0.29481191 + 0.84375 = 0.54893809 x = 0.54893809 Simplifying x = 0.54893809Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.13856191, 0.54893809}
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