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Simplifying 6x2 + (-10x) + 6 + 4x + (-5) = 0 6x2 + (-10x) + 6 + 4x + -5 = 0 Reorder the terms: 6 + -5 + (-10x) + 4x + 6x2 = 0 Combine like terms: 6 + -5 = 1 1 + (-10x) + 4x + 6x2 = 0 Combine like terms: (-10x) + 4x = -6x 1 + -6x + 6x2 = 0 Solving 1 + -6x + 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'. 0.1666666667 + -1x + x2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + -1x + -0.1666666667 + x2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + -1x + x2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + -1x + x2 = 0 + -0.1666666667 -1x + x2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 -1x + x2 = -0.1666666667 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.1666666667 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.1666666667 + 0.25 Combine like terms: -0.1666666667 + 0.25 = 0.0833333333 0.25 + -1x + x2 = 0.0833333333 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.0833333333 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (x + -0.5) equal to 0.288675135 and -0.288675135.Subproblem 1
x + -0.5 = 0.288675135 Simplifying x + -0.5 = 0.288675135 Reorder the terms: -0.5 + x = 0.288675135 Solving -0.5 + x = 0.288675135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.288675135 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.288675135 + 0.5 x = 0.288675135 + 0.5 Combine like terms: 0.288675135 + 0.5 = 0.788675135 x = 0.788675135 Simplifying x = 0.788675135Subproblem 2
x + -0.5 = -0.288675135 Simplifying x + -0.5 = -0.288675135 Reorder the terms: -0.5 + x = -0.288675135 Solving -0.5 + x = -0.288675135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.288675135 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.288675135 + 0.5 x = -0.288675135 + 0.5 Combine like terms: -0.288675135 + 0.5 = 0.211324865 x = 0.211324865 Simplifying x = 0.211324865Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.788675135, 0.211324865}
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