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Simplifying 6x2 + 9x = 32 Reorder the terms: 9x + 6x2 = 32 Solving 9x + 6x2 = 32 Solving for variable 'x'. Reorder the terms: -32 + 9x + 6x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 9x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.333333333 + 1.5x + x2 = 0 Move the constant term to the right: Add '5.333333333' to each side of the equation. -5.333333333 + 1.5x + 5.333333333 + x2 = 0 + 5.333333333 Reorder the terms: -5.333333333 + 5.333333333 + 1.5x + x2 = 0 + 5.333333333 Combine like terms: -5.333333333 + 5.333333333 = 0.000000000 0.000000000 + 1.5x + x2 = 0 + 5.333333333 1.5x + x2 = 0 + 5.333333333 Combine like terms: 0 + 5.333333333 = 5.333333333 1.5x + x2 = 5.333333333 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 = 5.333333333 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 5.333333333 + 0.5625 Combine like terms: 5.333333333 + 0.5625 = 5.895833333 0.5625 + 1.5x + x2 = 5.895833333 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 5.895833333 Calculate the square root of the right side: 2.428133714 Break this problem into two subproblems by setting (x + 0.75) equal to 2.428133714 and -2.428133714.Subproblem 1
x + 0.75 = 2.428133714 Simplifying x + 0.75 = 2.428133714 Reorder the terms: 0.75 + x = 2.428133714 Solving 0.75 + x = 2.428133714 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 = 2.428133714 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 2.428133714 + -0.75 x = 2.428133714 + -0.75 Combine like terms: 2.428133714 + -0.75 = 1.678133714 x = 1.678133714 Simplifying x = 1.678133714Subproblem 2
x + 0.75 = -2.428133714 Simplifying x + 0.75 = -2.428133714 Reorder the terms: 0.75 + x = -2.428133714 Solving 0.75 + x = -2.428133714 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 = -2.428133714 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -2.428133714 + -0.75 x = -2.428133714 + -0.75 Combine like terms: -2.428133714 + -0.75 = -3.178133714 x = -3.178133714 Simplifying x = -3.178133714Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.678133714, -3.178133714}
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