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Simplifying 16x2 + 24x + 9 = 12 Reorder the terms: 9 + 24x + 16x2 = 12 Solving 9 + 24x + 16x2 = 12 Solving for variable 'x'. Reorder the terms: 9 + -12 + 24x + 16x2 = 12 + -12 Combine like terms: 9 + -12 = -3 -3 + 24x + 16x2 = 12 + -12 Combine like terms: 12 + -12 = 0 -3 + 24x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + 1.5x + x2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + 1.5x + 0.1875 + x2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + 1.5x + x2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + 1.5x + x2 = 0 + 0.1875 1.5x + x2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 1.5x + x2 = 0.1875 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 0.1875 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 0.1875 + 0.5625 Combine like terms: 0.1875 + 0.5625 = 0.75 0.5625 + 1.5x + x2 = 0.75 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 0.75) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 0.75 = 0.866025404 Simplifying x + 0.75 = 0.866025404 Reorder the terms: 0.75 + x = 0.866025404 Solving 0.75 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 0.866025404 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 0.866025404 + -0.75 x = 0.866025404 + -0.75 Combine like terms: 0.866025404 + -0.75 = 0.116025404 x = 0.116025404 Simplifying x = 0.116025404Subproblem 2
x + 0.75 = -0.866025404 Simplifying x + 0.75 = -0.866025404 Reorder the terms: 0.75 + x = -0.866025404 Solving 0.75 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -0.866025404 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -0.866025404 + -0.75 x = -0.866025404 + -0.75 Combine like terms: -0.866025404 + -0.75 = -1.616025404 x = -1.616025404 Simplifying x = -1.616025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.116025404, -1.616025404}
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