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Simplifying 7x(x + 1) + -5 = 0 Reorder the terms: 7x(1 + x) + -5 = 0 (1 * 7x + x * 7x) + -5 = 0 (7x + 7x2) + -5 = 0 Reorder the terms: -5 + 7x + 7x2 = 0 Solving -5 + 7x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.7142857143 + x + x2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + x + 0.7142857143 + x2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + x + x2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + x + x2 = 0 + 0.7142857143 x + x2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 x + x2 = 0.7142857143 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 = 0.7142857143 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.7142857143 + 0.25 Combine like terms: 0.7142857143 + 0.25 = 0.9642857143 1.25 + x2 = 0.9642857143 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.9642857143 Calculate the square root of the right side: 0.981980506 Break this problem into two subproblems by setting (x + 0.5) equal to 0.981980506 and -0.981980506.Subproblem 1
x + 0.5 = 0.981980506 Simplifying x + 0.5 = 0.981980506 Reorder the terms: 0.5 + x = 0.981980506 Solving 0.5 + x = 0.981980506 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 = 0.981980506 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.981980506 + -0.5 x = 0.981980506 + -0.5 Combine like terms: 0.981980506 + -0.5 = 0.481980506 x = 0.481980506 Simplifying x = 0.481980506Subproblem 2
x + 0.5 = -0.981980506 Simplifying x + 0.5 = -0.981980506 Reorder the terms: 0.5 + x = -0.981980506 Solving 0.5 + x = -0.981980506 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 = -0.981980506 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.981980506 + -0.5 x = -0.981980506 + -0.5 Combine like terms: -0.981980506 + -0.5 = -1.481980506 x = -1.481980506 Simplifying x = -1.481980506Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.481980506, -1.481980506}
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