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Simplifying 40 = x2 + 2.5x + 14.5 Reorder the terms: 40 = 14.5 + 2.5x + x2 Solving 40 = 14.5 + 2.5x + x2 Solving for variable 'x'. Combine like terms: 40 + -14.5 = 25.5 25.5 + -2.5x + -1x2 = 14.5 + 2.5x + x2 + -14.5 + -2.5x + -1x2 Reorder the terms: 25.5 + -2.5x + -1x2 = 14.5 + -14.5 + 2.5x + -2.5x + x2 + -1x2 Combine like terms: 14.5 + -14.5 = 0.0 25.5 + -2.5x + -1x2 = 0.0 + 2.5x + -2.5x + x2 + -1x2 25.5 + -2.5x + -1x2 = 2.5x + -2.5x + x2 + -1x2 Combine like terms: 2.5x + -2.5x = 0.0 25.5 + -2.5x + -1x2 = 0.0 + x2 + -1x2 25.5 + -2.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 25.5 + -2.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -25.5 + 2.5x + x2 = 0 Move the constant term to the right: Add '25.5' to each side of the equation. -25.5 + 2.5x + 25.5 + x2 = 0 + 25.5 Reorder the terms: -25.5 + 25.5 + 2.5x + x2 = 0 + 25.5 Combine like terms: -25.5 + 25.5 = 0.0 0.0 + 2.5x + x2 = 0 + 25.5 2.5x + x2 = 0 + 25.5 Combine like terms: 0 + 25.5 = 25.5 2.5x + x2 = 25.5 The x term is 2.5x. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5x + 1.5625 + x2 = 25.5 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = 25.5 + 1.5625 Combine like terms: 25.5 + 1.5625 = 27.0625 1.5625 + 2.5x + x2 = 27.0625 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 27.0625 Calculate the square root of the right side: 5.202163012 Break this problem into two subproblems by setting (x + 1.25) equal to 5.202163012 and -5.202163012.Subproblem 1
x + 1.25 = 5.202163012 Simplifying x + 1.25 = 5.202163012 Reorder the terms: 1.25 + x = 5.202163012 Solving 1.25 + x = 5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 5.202163012 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 5.202163012 + -1.25 x = 5.202163012 + -1.25 Combine like terms: 5.202163012 + -1.25 = 3.952163012 x = 3.952163012 Simplifying x = 3.952163012Subproblem 2
x + 1.25 = -5.202163012 Simplifying x + 1.25 = -5.202163012 Reorder the terms: 1.25 + x = -5.202163012 Solving 1.25 + x = -5.202163012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -5.202163012 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -5.202163012 + -1.25 x = -5.202163012 + -1.25 Combine like terms: -5.202163012 + -1.25 = -6.452163012 x = -6.452163012 Simplifying x = -6.452163012Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.952163012, -6.452163012}
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