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Simplifying 7 + x(x + 1) = 30 Reorder the terms: 7 + x(1 + x) = 30 7 + (1 * x + x * x) = 30 7 + (1x + x2) = 30 Solving 7 + 1x + x2 = 30 Solving for variable 'x'. Reorder the terms: 7 + -30 + 1x + x2 = 30 + -30 Combine like terms: 7 + -30 = -23 -23 + 1x + x2 = 30 + -30 Combine like terms: 30 + -30 = 0 -23 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 1x + 23 + x2 = 0 + 23 Reorder the terms: -23 + 23 + 1x + x2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 1x + x2 = 0 + 23 1x + x2 = 0 + 23 Combine like terms: 0 + 23 = 23 1x + x2 = 23 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. x + 0.25 + x2 = 23 + 0.25 Reorder the terms: 0.25 + x + x2 = 23 + 0.25 Combine like terms: 23 + 0.25 = 23.25 0.25 + x + x2 = 23.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 23.25 Calculate the square root of the right side: 4.82182538 Break this problem into two subproblems by setting (x + 0.5) equal to 4.82182538 and -4.82182538.Subproblem 1
x + 0.5 = 4.82182538 Simplifying x + 0.5 = 4.82182538 Reorder the terms: 0.5 + x = 4.82182538 Solving 0.5 + x = 4.82182538 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 = 4.82182538 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.82182538 + -0.5 x = 4.82182538 + -0.5 Combine like terms: 4.82182538 + -0.5 = 4.32182538 x = 4.32182538 Simplifying x = 4.32182538Subproblem 2
x + 0.5 = -4.82182538 Simplifying x + 0.5 = -4.82182538 Reorder the terms: 0.5 + x = -4.82182538 Solving 0.5 + x = -4.82182538 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 = -4.82182538 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.82182538 + -0.5 x = -4.82182538 + -0.5 Combine like terms: -4.82182538 + -0.5 = -5.32182538 x = -5.32182538 Simplifying x = -5.32182538Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.32182538, -5.32182538}
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