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Simplifying 10x2 + 14x = -1x2 + 3x + 5 Reorder the terms: 14x + 10x2 = -1x2 + 3x + 5 Reorder the terms: 14x + 10x2 = 5 + 3x + -1x2 Solving 14x + 10x2 = 5 + 3x + -1x2 Solving for variable 'x'. Reorder the terms: -5 + 14x + -3x + 10x2 + x2 = 5 + 3x + -1x2 + -5 + -3x + x2 Combine like terms: 14x + -3x = 11x -5 + 11x + 10x2 + x2 = 5 + 3x + -1x2 + -5 + -3x + x2 Combine like terms: 10x2 + x2 = 11x2 -5 + 11x + 11x2 = 5 + 3x + -1x2 + -5 + -3x + x2 Reorder the terms: -5 + 11x + 11x2 = 5 + -5 + 3x + -3x + -1x2 + x2 Combine like terms: 5 + -5 = 0 -5 + 11x + 11x2 = 0 + 3x + -3x + -1x2 + x2 -5 + 11x + 11x2 = 3x + -3x + -1x2 + x2 Combine like terms: 3x + -3x = 0 -5 + 11x + 11x2 = 0 + -1x2 + x2 -5 + 11x + 11x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 -5 + 11x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.4545454545 + x + x2 = 0 Move the constant term to the right: Add '0.4545454545' to each side of the equation. -0.4545454545 + x + 0.4545454545 + x2 = 0 + 0.4545454545 Reorder the terms: -0.4545454545 + 0.4545454545 + x + x2 = 0 + 0.4545454545 Combine like terms: -0.4545454545 + 0.4545454545 = 0.0000000000 0.0000000000 + x + x2 = 0 + 0.4545454545 x + x2 = 0 + 0.4545454545 Combine like terms: 0 + 0.4545454545 = 0.4545454545 x + x2 = 0.4545454545 The x term is x. 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. + 0.25 + x2 = 0.4545454545 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.4545454545 + 0.25 Combine like terms: 0.4545454545 + 0.25 = 0.7045454545 1.25 + x2 = 0.7045454545 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.7045454545 Calculate the square root of the right side: 0.83937206 Break this problem into two subproblems by setting (x + 0.5) equal to 0.83937206 and -0.83937206.Subproblem 1
x + 0.5 = 0.83937206 Simplifying x + 0.5 = 0.83937206 Reorder the terms: 0.5 + x = 0.83937206 Solving 0.5 + x = 0.83937206 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 = 0.83937206 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.83937206 + -0.5 x = 0.83937206 + -0.5 Combine like terms: 0.83937206 + -0.5 = 0.33937206 x = 0.33937206 Simplifying x = 0.33937206Subproblem 2
x + 0.5 = -0.83937206 Simplifying x + 0.5 = -0.83937206 Reorder the terms: 0.5 + x = -0.83937206 Solving 0.5 + x = -0.83937206 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 = -0.83937206 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.83937206 + -0.5 x = -0.83937206 + -0.5 Combine like terms: -0.83937206 + -0.5 = -1.33937206 x = -1.33937206 Simplifying x = -1.33937206Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.33937206, -1.33937206}
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