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Simplifying 16x2 + 6x + 9 = 225 Reorder the terms: 9 + 6x + 16x2 = 225 Solving 9 + 6x + 16x2 = 225 Solving for variable 'x'. Reorder the terms: 9 + -225 + 6x + 16x2 = 225 + -225 Combine like terms: 9 + -225 = -216 -216 + 6x + 16x2 = 225 + -225 Combine like terms: 225 + -225 = 0 -216 + 6x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-108 + 3x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-108 + 3x + 8x2)' equal to zero and attempt to solve: Simplifying -108 + 3x + 8x2 = 0 Solving -108 + 3x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -13.5 + 0.375x + x2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + 0.375x + 13.5 + x2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + 0.375x + x2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + 0.375x + x2 = 0 + 13.5 0.375x + x2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 0.375x + x2 = 13.5 The x term is 0.375x. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375x + 0.03515625 + x2 = 13.5 + 0.03515625 Reorder the terms: 0.03515625 + 0.375x + x2 = 13.5 + 0.03515625 Combine like terms: 13.5 + 0.03515625 = 13.53515625 0.03515625 + 0.375x + x2 = 13.53515625 Factor a perfect square on the left side: (x + 0.1875)(x + 0.1875) = 13.53515625 Calculate the square root of the right side: 3.679015663 Break this problem into two subproblems by setting (x + 0.1875) equal to 3.679015663 and -3.679015663.Subproblem 1
x + 0.1875 = 3.679015663 Simplifying x + 0.1875 = 3.679015663 Reorder the terms: 0.1875 + x = 3.679015663 Solving 0.1875 + x = 3.679015663 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + x = 3.679015663 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = 3.679015663 + -0.1875 x = 3.679015663 + -0.1875 Combine like terms: 3.679015663 + -0.1875 = 3.491515663 x = 3.491515663 Simplifying x = 3.491515663Subproblem 2
x + 0.1875 = -3.679015663 Simplifying x + 0.1875 = -3.679015663 Reorder the terms: 0.1875 + x = -3.679015663 Solving 0.1875 + x = -3.679015663 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + x = -3.679015663 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = -3.679015663 + -0.1875 x = -3.679015663 + -0.1875 Combine like terms: -3.679015663 + -0.1875 = -3.866515663 x = -3.866515663 Simplifying x = -3.866515663Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.491515663, -3.866515663}Solution
x = {3.491515663, -3.866515663}
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