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Simplifying -16x2 + 86x + -53 = 0 Reorder the terms: -53 + 86x + -16x2 = 0 Solving -53 + 86x + -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'. 3.3125 + -5.375x + x2 = 0 Move the constant term to the right: Add '-3.3125' to each side of the equation. 3.3125 + -5.375x + -3.3125 + x2 = 0 + -3.3125 Reorder the terms: 3.3125 + -3.3125 + -5.375x + x2 = 0 + -3.3125 Combine like terms: 3.3125 + -3.3125 = 0.0000 0.0000 + -5.375x + x2 = 0 + -3.3125 -5.375x + x2 = 0 + -3.3125 Combine like terms: 0 + -3.3125 = -3.3125 -5.375x + x2 = -3.3125 The x term is -5.375x. Take half its coefficient (-2.6875). Square it (7.22265625) and add it to both sides. Add '7.22265625' to each side of the equation. -5.375x + 7.22265625 + x2 = -3.3125 + 7.22265625 Reorder the terms: 7.22265625 + -5.375x + x2 = -3.3125 + 7.22265625 Combine like terms: -3.3125 + 7.22265625 = 3.91015625 7.22265625 + -5.375x + x2 = 3.91015625 Factor a perfect square on the left side: (x + -2.6875)(x + -2.6875) = 3.91015625 Calculate the square root of the right side: 1.977411502 Break this problem into two subproblems by setting (x + -2.6875) equal to 1.977411502 and -1.977411502.Subproblem 1
x + -2.6875 = 1.977411502 Simplifying x + -2.6875 = 1.977411502 Reorder the terms: -2.6875 + x = 1.977411502 Solving -2.6875 + x = 1.977411502 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.6875' to each side of the equation. -2.6875 + 2.6875 + x = 1.977411502 + 2.6875 Combine like terms: -2.6875 + 2.6875 = 0.0000 0.0000 + x = 1.977411502 + 2.6875 x = 1.977411502 + 2.6875 Combine like terms: 1.977411502 + 2.6875 = 4.664911502 x = 4.664911502 Simplifying x = 4.664911502Subproblem 2
x + -2.6875 = -1.977411502 Simplifying x + -2.6875 = -1.977411502 Reorder the terms: -2.6875 + x = -1.977411502 Solving -2.6875 + x = -1.977411502 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.6875' to each side of the equation. -2.6875 + 2.6875 + x = -1.977411502 + 2.6875 Combine like terms: -2.6875 + 2.6875 = 0.0000 0.0000 + x = -1.977411502 + 2.6875 x = -1.977411502 + 2.6875 Combine like terms: -1.977411502 + 2.6875 = 0.710088498 x = 0.710088498 Simplifying x = 0.710088498Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.664911502, 0.710088498}
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