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Simplifying x(2 + x) + x + 2 = 78 (2 * x + x * x) + x + 2 = 78 (2x + x2) + x + 2 = 78 Reorder the terms: 2 + 2x + x + x2 = 78 Combine like terms: 2x + x = 3x 2 + 3x + x2 = 78 Solving 2 + 3x + x2 = 78 Solving for variable 'x'. Reorder the terms: 2 + -78 + 3x + x2 = 78 + -78 Combine like terms: 2 + -78 = -76 -76 + 3x + x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -76 + 3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '76' to each side of the equation. -76 + 3x + 76 + x2 = 0 + 76 Reorder the terms: -76 + 76 + 3x + x2 = 0 + 76 Combine like terms: -76 + 76 = 0 0 + 3x + x2 = 0 + 76 3x + x2 = 0 + 76 Combine like terms: 0 + 76 = 76 3x + x2 = 76 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 76 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 76 + 2.25 Combine like terms: 76 + 2.25 = 78.25 2.25 + 3x + x2 = 78.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 78.25 Calculate the square root of the right side: 8.845903006 Break this problem into two subproblems by setting (x + 1.5) equal to 8.845903006 and -8.845903006.Subproblem 1
x + 1.5 = 8.845903006 Simplifying x + 1.5 = 8.845903006 Reorder the terms: 1.5 + x = 8.845903006 Solving 1.5 + x = 8.845903006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 8.845903006 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 8.845903006 + -1.5 x = 8.845903006 + -1.5 Combine like terms: 8.845903006 + -1.5 = 7.345903006 x = 7.345903006 Simplifying x = 7.345903006Subproblem 2
x + 1.5 = -8.845903006 Simplifying x + 1.5 = -8.845903006 Reorder the terms: 1.5 + x = -8.845903006 Solving 1.5 + x = -8.845903006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -8.845903006 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -8.845903006 + -1.5 x = -8.845903006 + -1.5 Combine like terms: -8.845903006 + -1.5 = -10.345903006 x = -10.345903006 Simplifying x = -10.345903006Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.345903006, -10.345903006}
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