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Simplifying -16x2 + 7.8x + 12 = 0 Reorder the terms: 12 + 7.8x + -16x2 = 0 Solving 12 + 7.8x + -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'. -0.75 + -0.4875x + x2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.4875x + 0.75 + x2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.4875x + x2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.4875x + x2 = 0 + 0.75 -0.4875x + x2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.4875x + x2 = 0.75 The x term is -0.4875x. Take half its coefficient (-0.24375). Square it (0.0594140625) and add it to both sides. Add '0.0594140625' to each side of the equation. -0.4875x + 0.0594140625 + x2 = 0.75 + 0.0594140625 Reorder the terms: 0.0594140625 + -0.4875x + x2 = 0.75 + 0.0594140625 Combine like terms: 0.75 + 0.0594140625 = 0.8094140625 0.0594140625 + -0.4875x + x2 = 0.8094140625 Factor a perfect square on the left side: (x + -0.24375)(x + -0.24375) = 0.8094140625 Calculate the square root of the right side: 0.89967442 Break this problem into two subproblems by setting (x + -0.24375) equal to 0.89967442 and -0.89967442.Subproblem 1
x + -0.24375 = 0.89967442 Simplifying x + -0.24375 = 0.89967442 Reorder the terms: -0.24375 + x = 0.89967442 Solving -0.24375 + x = 0.89967442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.24375' to each side of the equation. -0.24375 + 0.24375 + x = 0.89967442 + 0.24375 Combine like terms: -0.24375 + 0.24375 = 0.00000 0.00000 + x = 0.89967442 + 0.24375 x = 0.89967442 + 0.24375 Combine like terms: 0.89967442 + 0.24375 = 1.14342442 x = 1.14342442 Simplifying x = 1.14342442Subproblem 2
x + -0.24375 = -0.89967442 Simplifying x + -0.24375 = -0.89967442 Reorder the terms: -0.24375 + x = -0.89967442 Solving -0.24375 + x = -0.89967442 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.24375' to each side of the equation. -0.24375 + 0.24375 + x = -0.89967442 + 0.24375 Combine like terms: -0.24375 + 0.24375 = 0.00000 0.00000 + x = -0.89967442 + 0.24375 x = -0.89967442 + 0.24375 Combine like terms: -0.89967442 + 0.24375 = -0.65592442 x = -0.65592442 Simplifying x = -0.65592442Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.14342442, -0.65592442}
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