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Simplifying 16x2 + -7x + 9 + -31x = 0 Reorder the terms: 9 + -7x + -31x + 16x2 = 0 Combine like terms: -7x + -31x = -38x 9 + -38x + 16x2 = 0 Solving 9 + -38x + 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'. 0.5625 + -2.375x + x2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + -2.375x + -0.5625 + x2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + -2.375x + x2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + -2.375x + x2 = 0 + -0.5625 -2.375x + x2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 -2.375x + x2 = -0.5625 The x term is -2.375x. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375x + 1.41015625 + x2 = -0.5625 + 1.41015625 Reorder the terms: 1.41015625 + -2.375x + x2 = -0.5625 + 1.41015625 Combine like terms: -0.5625 + 1.41015625 = 0.84765625 1.41015625 + -2.375x + x2 = 0.84765625 Factor a perfect square on the left side: (x + -1.1875)(x + -1.1875) = 0.84765625 Calculate the square root of the right side: 0.920682491 Break this problem into two subproblems by setting (x + -1.1875) equal to 0.920682491 and -0.920682491.Subproblem 1
x + -1.1875 = 0.920682491 Simplifying x + -1.1875 = 0.920682491 Reorder the terms: -1.1875 + x = 0.920682491 Solving -1.1875 + x = 0.920682491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + x = 0.920682491 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + x = 0.920682491 + 1.1875 x = 0.920682491 + 1.1875 Combine like terms: 0.920682491 + 1.1875 = 2.108182491 x = 2.108182491 Simplifying x = 2.108182491Subproblem 2
x + -1.1875 = -0.920682491 Simplifying x + -1.1875 = -0.920682491 Reorder the terms: -1.1875 + x = -0.920682491 Solving -1.1875 + x = -0.920682491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + x = -0.920682491 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + x = -0.920682491 + 1.1875 x = -0.920682491 + 1.1875 Combine like terms: -0.920682491 + 1.1875 = 0.266817509 x = 0.266817509 Simplifying x = 0.266817509Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.108182491, 0.266817509}
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