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Simplifying (x + 7)(2x + -3) = 140 Reorder the terms: (7 + x)(2x + -3) = 140 Reorder the terms: (7 + x)(-3 + 2x) = 140 Multiply (7 + x) * (-3 + 2x) (7(-3 + 2x) + x(-3 + 2x)) = 140 ((-3 * 7 + 2x * 7) + x(-3 + 2x)) = 140 ((-21 + 14x) + x(-3 + 2x)) = 140 (-21 + 14x + (-3 * x + 2x * x)) = 140 (-21 + 14x + (-3x + 2x2)) = 140 Combine like terms: 14x + -3x = 11x (-21 + 11x + 2x2) = 140 Solving -21 + 11x + 2x2 = 140 Solving for variable 'x'. Reorder the terms: -21 + -140 + 11x + 2x2 = 140 + -140 Combine like terms: -21 + -140 = -161 -161 + 11x + 2x2 = 140 + -140 Combine like terms: 140 + -140 = 0 -161 + 11x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -80.5 + 5.5x + x2 = 0 Move the constant term to the right: Add '80.5' to each side of the equation. -80.5 + 5.5x + 80.5 + x2 = 0 + 80.5 Reorder the terms: -80.5 + 80.5 + 5.5x + x2 = 0 + 80.5 Combine like terms: -80.5 + 80.5 = 0.0 0.0 + 5.5x + x2 = 0 + 80.5 5.5x + x2 = 0 + 80.5 Combine like terms: 0 + 80.5 = 80.5 5.5x + x2 = 80.5 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 = 80.5 + 7.5625 Reorder the terms: 7.5625 + 5.5x + x2 = 80.5 + 7.5625 Combine like terms: 80.5 + 7.5625 = 88.0625 7.5625 + 5.5x + x2 = 88.0625 Factor a perfect square on the left side: (x + 2.75)(x + 2.75) = 88.0625 Calculate the square root of the right side: 9.38416219 Break this problem into two subproblems by setting (x + 2.75) equal to 9.38416219 and -9.38416219.Subproblem 1
x + 2.75 = 9.38416219 Simplifying x + 2.75 = 9.38416219 Reorder the terms: 2.75 + x = 9.38416219 Solving 2.75 + x = 9.38416219 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 = 9.38416219 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 9.38416219 + -2.75 x = 9.38416219 + -2.75 Combine like terms: 9.38416219 + -2.75 = 6.63416219 x = 6.63416219 Simplifying x = 6.63416219Subproblem 2
x + 2.75 = -9.38416219 Simplifying x + 2.75 = -9.38416219 Reorder the terms: 2.75 + x = -9.38416219 Solving 2.75 + x = -9.38416219 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 = -9.38416219 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -9.38416219 + -2.75 x = -9.38416219 + -2.75 Combine like terms: -9.38416219 + -2.75 = -12.13416219 x = -12.13416219 Simplifying x = -12.13416219Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.63416219, -12.13416219}
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