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Simplifying 16x2 + 78x + -27 = 0 Reorder the terms: -27 + 78x + 16x2 = 0 Solving -27 + 78x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.6875 + 4.875x + x2 = 0 Move the constant term to the right: Add '1.6875' to each side of the equation. -1.6875 + 4.875x + 1.6875 + x2 = 0 + 1.6875 Reorder the terms: -1.6875 + 1.6875 + 4.875x + x2 = 0 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + 4.875x + x2 = 0 + 1.6875 4.875x + x2 = 0 + 1.6875 Combine like terms: 0 + 1.6875 = 1.6875 4.875x + x2 = 1.6875 The x term is 4.875x. Take half its coefficient (2.4375). Square it (5.94140625) and add it to both sides. Add '5.94140625' to each side of the equation. 4.875x + 5.94140625 + x2 = 1.6875 + 5.94140625 Reorder the terms: 5.94140625 + 4.875x + x2 = 1.6875 + 5.94140625 Combine like terms: 1.6875 + 5.94140625 = 7.62890625 5.94140625 + 4.875x + x2 = 7.62890625 Factor a perfect square on the left side: (x + 2.4375)(x + 2.4375) = 7.62890625 Calculate the square root of the right side: 2.762047474 Break this problem into two subproblems by setting (x + 2.4375) equal to 2.762047474 and -2.762047474.Subproblem 1
x + 2.4375 = 2.762047474 Simplifying x + 2.4375 = 2.762047474 Reorder the terms: 2.4375 + x = 2.762047474 Solving 2.4375 + x = 2.762047474 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = 2.762047474 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = 2.762047474 + -2.4375 x = 2.762047474 + -2.4375 Combine like terms: 2.762047474 + -2.4375 = 0.324547474 x = 0.324547474 Simplifying x = 0.324547474Subproblem 2
x + 2.4375 = -2.762047474 Simplifying x + 2.4375 = -2.762047474 Reorder the terms: 2.4375 + x = -2.762047474 Solving 2.4375 + x = -2.762047474 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.4375' to each side of the equation. 2.4375 + -2.4375 + x = -2.762047474 + -2.4375 Combine like terms: 2.4375 + -2.4375 = 0.0000 0.0000 + x = -2.762047474 + -2.4375 x = -2.762047474 + -2.4375 Combine like terms: -2.762047474 + -2.4375 = -5.199547474 x = -5.199547474 Simplifying x = -5.199547474Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.324547474, -5.199547474}
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