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Simplifying (5 + 2x)(x + 6) = 213 Reorder the terms: (5 + 2x)(6 + x) = 213 Multiply (5 + 2x) * (6 + x) (5(6 + x) + 2x * (6 + x)) = 213 ((6 * 5 + x * 5) + 2x * (6 + x)) = 213 ((30 + 5x) + 2x * (6 + x)) = 213 (30 + 5x + (6 * 2x + x * 2x)) = 213 (30 + 5x + (12x + 2x2)) = 213 Combine like terms: 5x + 12x = 17x (30 + 17x + 2x2) = 213 Solving 30 + 17x + 2x2 = 213 Solving for variable 'x'. Reorder the terms: 30 + -213 + 17x + 2x2 = 213 + -213 Combine like terms: 30 + -213 = -183 -183 + 17x + 2x2 = 213 + -213 Combine like terms: 213 + -213 = 0 -183 + 17x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -91.5 + 8.5x + x2 = 0 Move the constant term to the right: Add '91.5' to each side of the equation. -91.5 + 8.5x + 91.5 + x2 = 0 + 91.5 Reorder the terms: -91.5 + 91.5 + 8.5x + x2 = 0 + 91.5 Combine like terms: -91.5 + 91.5 = 0.0 0.0 + 8.5x + x2 = 0 + 91.5 8.5x + x2 = 0 + 91.5 Combine like terms: 0 + 91.5 = 91.5 8.5x + x2 = 91.5 The x term is 8.5x. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5x + 18.0625 + x2 = 91.5 + 18.0625 Reorder the terms: 18.0625 + 8.5x + x2 = 91.5 + 18.0625 Combine like terms: 91.5 + 18.0625 = 109.5625 18.0625 + 8.5x + x2 = 109.5625 Factor a perfect square on the left side: (x + 4.25)(x + 4.25) = 109.5625 Calculate the square root of the right side: 10.467210708 Break this problem into two subproblems by setting (x + 4.25) equal to 10.467210708 and -10.467210708.Subproblem 1
x + 4.25 = 10.467210708 Simplifying x + 4.25 = 10.467210708 Reorder the terms: 4.25 + x = 10.467210708 Solving 4.25 + x = 10.467210708 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 10.467210708 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 10.467210708 + -4.25 x = 10.467210708 + -4.25 Combine like terms: 10.467210708 + -4.25 = 6.217210708 x = 6.217210708 Simplifying x = 6.217210708Subproblem 2
x + 4.25 = -10.467210708 Simplifying x + 4.25 = -10.467210708 Reorder the terms: 4.25 + x = -10.467210708 Solving 4.25 + x = -10.467210708 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -10.467210708 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -10.467210708 + -4.25 x = -10.467210708 + -4.25 Combine like terms: -10.467210708 + -4.25 = -14.717210708 x = -14.717210708 Simplifying x = -14.717210708Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.217210708, -14.717210708}
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