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Simplifying 2x2 + 3x = 67 Reorder the terms: 3x + 2x2 = 67 Solving 3x + 2x2 = 67 Solving for variable 'x'. Reorder the terms: -67 + 3x + 2x2 = 67 + -67 Combine like terms: 67 + -67 = 0 -67 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -33.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '33.5' to each side of the equation. -33.5 + 1.5x + 33.5 + x2 = 0 + 33.5 Reorder the terms: -33.5 + 33.5 + 1.5x + x2 = 0 + 33.5 Combine like terms: -33.5 + 33.5 = 0.0 0.0 + 1.5x + x2 = 0 + 33.5 1.5x + x2 = 0 + 33.5 Combine like terms: 0 + 33.5 = 33.5 1.5x + x2 = 33.5 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 = 33.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 33.5 + 0.5625 Combine like terms: 33.5 + 0.5625 = 34.0625 0.5625 + 1.5x + x2 = 34.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 34.0625 Calculate the square root of the right side: 5.836308765 Break this problem into two subproblems by setting (x + 0.75) equal to 5.836308765 and -5.836308765.Subproblem 1
x + 0.75 = 5.836308765 Simplifying x + 0.75 = 5.836308765 Reorder the terms: 0.75 + x = 5.836308765 Solving 0.75 + x = 5.836308765 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 = 5.836308765 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 5.836308765 + -0.75 x = 5.836308765 + -0.75 Combine like terms: 5.836308765 + -0.75 = 5.086308765 x = 5.086308765 Simplifying x = 5.086308765Subproblem 2
x + 0.75 = -5.836308765 Simplifying x + 0.75 = -5.836308765 Reorder the terms: 0.75 + x = -5.836308765 Solving 0.75 + x = -5.836308765 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 = -5.836308765 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -5.836308765 + -0.75 x = -5.836308765 + -0.75 Combine like terms: -5.836308765 + -0.75 = -6.586308765 x = -6.586308765 Simplifying x = -6.586308765Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.086308765, -6.586308765}
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