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Simplifying 6x2 + 6.5x + -32 = 0 Reorder the terms: -32 + 6.5x + 6x2 = 0 Solving -32 + 6.5x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.333333333 + 1.083333333x + x2 = 0 Move the constant term to the right: Add '5.333333333' to each side of the equation. -5.333333333 + 1.083333333x + 5.333333333 + x2 = 0 + 5.333333333 Reorder the terms: -5.333333333 + 5.333333333 + 1.083333333x + x2 = 0 + 5.333333333 Combine like terms: -5.333333333 + 5.333333333 = 0.000000000 0.000000000 + 1.083333333x + x2 = 0 + 5.333333333 1.083333333x + x2 = 0 + 5.333333333 Combine like terms: 0 + 5.333333333 = 5.333333333 1.083333333x + x2 = 5.333333333 The x term is 1.083333333x. Take half its coefficient (0.5416666665). Square it (0.2934027776) and add it to both sides. Add '0.2934027776' to each side of the equation. 1.083333333x + 0.2934027776 + x2 = 5.333333333 + 0.2934027776 Reorder the terms: 0.2934027776 + 1.083333333x + x2 = 5.333333333 + 0.2934027776 Combine like terms: 5.333333333 + 0.2934027776 = 5.6267361106 0.2934027776 + 1.083333333x + x2 = 5.6267361106 Factor a perfect square on the left side: (x + 0.5416666665)(x + 0.5416666665) = 5.6267361106 Calculate the square root of the right side: 2.372074221 Break this problem into two subproblems by setting (x + 0.5416666665) equal to 2.372074221 and -2.372074221.Subproblem 1
x + 0.5416666665 = 2.372074221 Simplifying x + 0.5416666665 = 2.372074221 Reorder the terms: 0.5416666665 + x = 2.372074221 Solving 0.5416666665 + x = 2.372074221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5416666665' to each side of the equation. 0.5416666665 + -0.5416666665 + x = 2.372074221 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = 2.372074221 + -0.5416666665 x = 2.372074221 + -0.5416666665 Combine like terms: 2.372074221 + -0.5416666665 = 1.8304075545 x = 1.8304075545 Simplifying x = 1.8304075545Subproblem 2
x + 0.5416666665 = -2.372074221 Simplifying x + 0.5416666665 = -2.372074221 Reorder the terms: 0.5416666665 + x = -2.372074221 Solving 0.5416666665 + x = -2.372074221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5416666665' to each side of the equation. 0.5416666665 + -0.5416666665 + x = -2.372074221 + -0.5416666665 Combine like terms: 0.5416666665 + -0.5416666665 = 0.0000000000 0.0000000000 + x = -2.372074221 + -0.5416666665 x = -2.372074221 + -0.5416666665 Combine like terms: -2.372074221 + -0.5416666665 = -2.9137408875 x = -2.9137408875 Simplifying x = -2.9137408875Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.8304075545, -2.9137408875}
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