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Simplifying 16x2 + 10x + -460 = 0 Reorder the terms: -460 + 10x + 16x2 = 0 Solving -460 + 10x + 16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-230 + 5x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-230 + 5x + 8x2)' equal to zero and attempt to solve: Simplifying -230 + 5x + 8x2 = 0 Solving -230 + 5x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -28.75 + 0.625x + x2 = 0 Move the constant term to the right: Add '28.75' to each side of the equation. -28.75 + 0.625x + 28.75 + x2 = 0 + 28.75 Reorder the terms: -28.75 + 28.75 + 0.625x + x2 = 0 + 28.75 Combine like terms: -28.75 + 28.75 = 0.00 0.00 + 0.625x + x2 = 0 + 28.75 0.625x + x2 = 0 + 28.75 Combine like terms: 0 + 28.75 = 28.75 0.625x + x2 = 28.75 The x term is 0.625x. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625x + 0.09765625 + x2 = 28.75 + 0.09765625 Reorder the terms: 0.09765625 + 0.625x + x2 = 28.75 + 0.09765625 Combine like terms: 28.75 + 0.09765625 = 28.84765625 0.09765625 + 0.625x + x2 = 28.84765625 Factor a perfect square on the left side: (x + 0.3125)(x + 0.3125) = 28.84765625 Calculate the square root of the right side: 5.37100142 Break this problem into two subproblems by setting (x + 0.3125) equal to 5.37100142 and -5.37100142.Subproblem 1
x + 0.3125 = 5.37100142 Simplifying x + 0.3125 = 5.37100142 Reorder the terms: 0.3125 + x = 5.37100142 Solving 0.3125 + x = 5.37100142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + x = 5.37100142 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + x = 5.37100142 + -0.3125 x = 5.37100142 + -0.3125 Combine like terms: 5.37100142 + -0.3125 = 5.05850142 x = 5.05850142 Simplifying x = 5.05850142Subproblem 2
x + 0.3125 = -5.37100142 Simplifying x + 0.3125 = -5.37100142 Reorder the terms: 0.3125 + x = -5.37100142 Solving 0.3125 + x = -5.37100142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + x = -5.37100142 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + x = -5.37100142 + -0.3125 x = -5.37100142 + -0.3125 Combine like terms: -5.37100142 + -0.3125 = -5.68350142 x = -5.68350142 Simplifying x = -5.68350142Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.05850142, -5.68350142}Solution
x = {5.05850142, -5.68350142}
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