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Simplifying 0 = 8x2 + 27x + 14 Reorder the terms: 0 = 14 + 27x + 8x2 Solving 0 = 14 + 27x + 8x2 Solving for variable 'x'. Combine like terms: 0 + -14 = -14 -14 + -27x + -8x2 = 14 + 27x + 8x2 + -14 + -27x + -8x2 Reorder the terms: -14 + -27x + -8x2 = 14 + -14 + 27x + -27x + 8x2 + -8x2 Combine like terms: 14 + -14 = 0 -14 + -27x + -8x2 = 0 + 27x + -27x + 8x2 + -8x2 -14 + -27x + -8x2 = 27x + -27x + 8x2 + -8x2 Combine like terms: 27x + -27x = 0 -14 + -27x + -8x2 = 0 + 8x2 + -8x2 -14 + -27x + -8x2 = 8x2 + -8x2 Combine like terms: 8x2 + -8x2 = 0 -14 + -27x + -8x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(14 + 27x + 8x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(14 + 27x + 8x2)' equal to zero and attempt to solve: Simplifying 14 + 27x + 8x2 = 0 Solving 14 + 27x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.75 + 3.375x + x2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 3.375x + -1.75 + x2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 3.375x + x2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 3.375x + x2 = 0 + -1.75 3.375x + x2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 3.375x + x2 = -1.75 The x term is 3.375x. Take half its coefficient (1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. 3.375x + 2.84765625 + x2 = -1.75 + 2.84765625 Reorder the terms: 2.84765625 + 3.375x + x2 = -1.75 + 2.84765625 Combine like terms: -1.75 + 2.84765625 = 1.09765625 2.84765625 + 3.375x + x2 = 1.09765625 Factor a perfect square on the left side: (x + 1.6875)(x + 1.6875) = 1.09765625 Calculate the square root of the right side: 1.047690913 Break this problem into two subproblems by setting (x + 1.6875) equal to 1.047690913 and -1.047690913.Subproblem 1
x + 1.6875 = 1.047690913 Simplifying x + 1.6875 = 1.047690913 Reorder the terms: 1.6875 + x = 1.047690913 Solving 1.6875 + x = 1.047690913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + x = 1.047690913 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + x = 1.047690913 + -1.6875 x = 1.047690913 + -1.6875 Combine like terms: 1.047690913 + -1.6875 = -0.639809087 x = -0.639809087 Simplifying x = -0.639809087Subproblem 2
x + 1.6875 = -1.047690913 Simplifying x + 1.6875 = -1.047690913 Reorder the terms: 1.6875 + x = -1.047690913 Solving 1.6875 + x = -1.047690913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + x = -1.047690913 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + x = -1.047690913 + -1.6875 x = -1.047690913 + -1.6875 Combine like terms: -1.047690913 + -1.6875 = -2.735190913 x = -2.735190913 Simplifying x = -2.735190913Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.639809087, -2.735190913}Solution
x = {-0.639809087, -2.735190913}
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