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Simplifying (x + 5)(3x + -2) = 14 Reorder the terms: (5 + x)(3x + -2) = 14 Reorder the terms: (5 + x)(-2 + 3x) = 14 Multiply (5 + x) * (-2 + 3x) (5(-2 + 3x) + x(-2 + 3x)) = 14 ((-2 * 5 + 3x * 5) + x(-2 + 3x)) = 14 ((-10 + 15x) + x(-2 + 3x)) = 14 (-10 + 15x + (-2 * x + 3x * x)) = 14 (-10 + 15x + (-2x + 3x2)) = 14 Combine like terms: 15x + -2x = 13x (-10 + 13x + 3x2) = 14 Solving -10 + 13x + 3x2 = 14 Solving for variable 'x'. Reorder the terms: -10 + -14 + 13x + 3x2 = 14 + -14 Combine like terms: -10 + -14 = -24 -24 + 13x + 3x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -24 + 13x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -8 + 4.333333333x + x2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 4.333333333x + 8 + x2 = 0 + 8 Reorder the terms: -8 + 8 + 4.333333333x + x2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 4.333333333x + x2 = 0 + 8 4.333333333x + x2 = 0 + 8 Combine like terms: 0 + 8 = 8 4.333333333x + x2 = 8 The x term is 4.333333333x. Take half its coefficient (2.166666667). Square it (4.694444446) and add it to both sides. Add '4.694444446' to each side of the equation. 4.333333333x + 4.694444446 + x2 = 8 + 4.694444446 Reorder the terms: 4.694444446 + 4.333333333x + x2 = 8 + 4.694444446 Combine like terms: 8 + 4.694444446 = 12.694444446 4.694444446 + 4.333333333x + x2 = 12.694444446 Factor a perfect square on the left side: (x + 2.166666667)(x + 2.166666667) = 12.694444446 Calculate the square root of the right side: 3.562926388 Break this problem into two subproblems by setting (x + 2.166666667) equal to 3.562926388 and -3.562926388.Subproblem 1
x + 2.166666667 = 3.562926388 Simplifying x + 2.166666667 = 3.562926388 Reorder the terms: 2.166666667 + x = 3.562926388 Solving 2.166666667 + x = 3.562926388 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + x = 3.562926388 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + x = 3.562926388 + -2.166666667 x = 3.562926388 + -2.166666667 Combine like terms: 3.562926388 + -2.166666667 = 1.396259721 x = 1.396259721 Simplifying x = 1.396259721Subproblem 2
x + 2.166666667 = -3.562926388 Simplifying x + 2.166666667 = -3.562926388 Reorder the terms: 2.166666667 + x = -3.562926388 Solving 2.166666667 + x = -3.562926388 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + x = -3.562926388 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + x = -3.562926388 + -2.166666667 x = -3.562926388 + -2.166666667 Combine like terms: -3.562926388 + -2.166666667 = -5.729593055 x = -5.729593055 Simplifying x = -5.729593055Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.396259721, -5.729593055}
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