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Simplifying 52 = x(x + 1) Reorder the terms: 52 = x(1 + x) 52 = (1 * x + x * x) 52 = (1x + x2) Solving 52 = 1x + x2 Solving for variable 'x'. Reorder the terms: 52 + -1x + -1x2 = 1x + -1x + x2 + -1x2 Combine like terms: 1x + -1x = 0 52 + -1x + -1x2 = 0 + x2 + -1x2 52 + -1x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 52 + -1x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -52 + x + x2 = 0 Move the constant term to the right: Add '52' to each side of the equation. -52 + x + 52 + x2 = 0 + 52 Reorder the terms: -52 + 52 + x + x2 = 0 + 52 Combine like terms: -52 + 52 = 0 0 + x + x2 = 0 + 52 x + x2 = 0 + 52 Combine like terms: 0 + 52 = 52 x + x2 = 52 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 = 52 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 52 + 0.25 Combine like terms: 52 + 0.25 = 52.25 1.25 + x2 = 52.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 52.25 Calculate the square root of the right side: 7.228416147 Break this problem into two subproblems by setting (x + 0.5) equal to 7.228416147 and -7.228416147.Subproblem 1
x + 0.5 = 7.228416147 Simplifying x + 0.5 = 7.228416147 Reorder the terms: 0.5 + x = 7.228416147 Solving 0.5 + x = 7.228416147 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 = 7.228416147 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.228416147 + -0.5 x = 7.228416147 + -0.5 Combine like terms: 7.228416147 + -0.5 = 6.728416147 x = 6.728416147 Simplifying x = 6.728416147Subproblem 2
x + 0.5 = -7.228416147 Simplifying x + 0.5 = -7.228416147 Reorder the terms: 0.5 + x = -7.228416147 Solving 0.5 + x = -7.228416147 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 = -7.228416147 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.228416147 + -0.5 x = -7.228416147 + -0.5 Combine like terms: -7.228416147 + -0.5 = -7.728416147 x = -7.728416147 Simplifying x = -7.728416147Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.728416147, -7.728416147}
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