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Simplifying (x + 3)(2x + 5) = 42 Reorder the terms: (3 + x)(2x + 5) = 42 Reorder the terms: (3 + x)(5 + 2x) = 42 Multiply (3 + x) * (5 + 2x) (3(5 + 2x) + x(5 + 2x)) = 42 ((5 * 3 + 2x * 3) + x(5 + 2x)) = 42 ((15 + 6x) + x(5 + 2x)) = 42 (15 + 6x + (5 * x + 2x * x)) = 42 (15 + 6x + (5x + 2x2)) = 42 Combine like terms: 6x + 5x = 11x (15 + 11x + 2x2) = 42 Solving 15 + 11x + 2x2 = 42 Solving for variable 'x'. Reorder the terms: 15 + -42 + 11x + 2x2 = 42 + -42 Combine like terms: 15 + -42 = -27 -27 + 11x + 2x2 = 42 + -42 Combine like terms: 42 + -42 = 0 -27 + 11x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -13.5 + 5.5x + x2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + 5.5x + 13.5 + x2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + 5.5x + x2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + 5.5x + x2 = 0 + 13.5 5.5x + x2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 5.5x + x2 = 13.5 The x term is 5.5x. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5x + 7.5625 + x2 = 13.5 + 7.5625 Reorder the terms: 7.5625 + 5.5x + x2 = 13.5 + 7.5625 Combine like terms: 13.5 + 7.5625 = 21.0625 7.5625 + 5.5x + x2 = 21.0625 Factor a perfect square on the left side: (x + 2.75)(x + 2.75) = 21.0625 Calculate the square root of the right side: 4.589389938 Break this problem into two subproblems by setting (x + 2.75) equal to 4.589389938 and -4.589389938.Subproblem 1
x + 2.75 = 4.589389938 Simplifying x + 2.75 = 4.589389938 Reorder the terms: 2.75 + x = 4.589389938 Solving 2.75 + x = 4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 4.589389938 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 4.589389938 + -2.75 x = 4.589389938 + -2.75 Combine like terms: 4.589389938 + -2.75 = 1.839389938 x = 1.839389938 Simplifying x = 1.839389938Subproblem 2
x + 2.75 = -4.589389938 Simplifying x + 2.75 = -4.589389938 Reorder the terms: 2.75 + x = -4.589389938 Solving 2.75 + x = -4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -4.589389938 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -4.589389938 + -2.75 x = -4.589389938 + -2.75 Combine like terms: -4.589389938 + -2.75 = -7.339389938 x = -7.339389938 Simplifying x = -7.339389938Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.839389938, -7.339389938}
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