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Simplifying x(x + 1) = 78 Reorder the terms: x(1 + x) = 78 (1 * x + x * x) = 78 (1x + x2) = 78 Solving 1x + x2 = 78 Solving for variable 'x'. Reorder the terms: -78 + 1x + x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -78 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 1x + 78 + x2 = 0 + 78 Reorder the terms: -78 + 78 + 1x + x2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 1x + x2 = 0 + 78 1x + x2 = 0 + 78 Combine like terms: 0 + 78 = 78 1x + x2 = 78 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 = 78 + 0.25 Reorder the terms: 0.25 + x + x2 = 78 + 0.25 Combine like terms: 78 + 0.25 = 78.25 0.25 + x + x2 = 78.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 78.25 Calculate the square root of the right side: 8.845903006 Break this problem into two subproblems by setting (x + 0.5) equal to 8.845903006 and -8.845903006.Subproblem 1
x + 0.5 = 8.845903006 Simplifying x + 0.5 = 8.845903006 Reorder the terms: 0.5 + x = 8.845903006 Solving 0.5 + x = 8.845903006 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 = 8.845903006 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.845903006 + -0.5 x = 8.845903006 + -0.5 Combine like terms: 8.845903006 + -0.5 = 8.345903006 x = 8.345903006 Simplifying x = 8.345903006Subproblem 2
x + 0.5 = -8.845903006 Simplifying x + 0.5 = -8.845903006 Reorder the terms: 0.5 + x = -8.845903006 Solving 0.5 + x = -8.845903006 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 = -8.845903006 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.845903006 + -0.5 x = -8.845903006 + -0.5 Combine like terms: -8.845903006 + -0.5 = -9.345903006 x = -9.345903006 Simplifying x = -9.345903006Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.345903006, -9.345903006}
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