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Simplifying (x + 3)(3x + -2) = 24 Reorder the terms: (3 + x)(3x + -2) = 24 Reorder the terms: (3 + x)(-2 + 3x) = 24 Multiply (3 + x) * (-2 + 3x) (3(-2 + 3x) + x(-2 + 3x)) = 24 ((-2 * 3 + 3x * 3) + x(-2 + 3x)) = 24 ((-6 + 9x) + x(-2 + 3x)) = 24 (-6 + 9x + (-2 * x + 3x * x)) = 24 (-6 + 9x + (-2x + 3x2)) = 24 Combine like terms: 9x + -2x = 7x (-6 + 7x + 3x2) = 24 Solving -6 + 7x + 3x2 = 24 Solving for variable 'x'. Reorder the terms: -6 + -24 + 7x + 3x2 = 24 + -24 Combine like terms: -6 + -24 = -30 -30 + 7x + 3x2 = 24 + -24 Combine like terms: 24 + -24 = 0 -30 + 7x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -10 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 2.333333333x + 10 + x2 = 0 + 10 Reorder the terms: -10 + 10 + 2.333333333x + x2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 2.333333333x + x2 = 0 + 10 2.333333333x + x2 = 0 + 10 Combine like terms: 0 + 10 = 10 2.333333333x + x2 = 10 The x term is 2.333333333x. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333x + 1.361111112 + x2 = 10 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 10 + 1.361111112 Combine like terms: 10 + 1.361111112 = 11.361111112 1.361111112 + 2.333333333x + x2 = 11.361111112 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 11.361111112 Calculate the square root of the right side: 3.370624736 Break this problem into two subproblems by setting (x + 1.166666667) equal to 3.370624736 and -3.370624736.Subproblem 1
x + 1.166666667 = 3.370624736 Simplifying x + 1.166666667 = 3.370624736 Reorder the terms: 1.166666667 + x = 3.370624736 Solving 1.166666667 + x = 3.370624736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 3.370624736 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 3.370624736 + -1.166666667 x = 3.370624736 + -1.166666667 Combine like terms: 3.370624736 + -1.166666667 = 2.203958069 x = 2.203958069 Simplifying x = 2.203958069Subproblem 2
x + 1.166666667 = -3.370624736 Simplifying x + 1.166666667 = -3.370624736 Reorder the terms: 1.166666667 + x = -3.370624736 Solving 1.166666667 + x = -3.370624736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -3.370624736 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -3.370624736 + -1.166666667 x = -3.370624736 + -1.166666667 Combine like terms: -3.370624736 + -1.166666667 = -4.537291403 x = -4.537291403 Simplifying x = -4.537291403Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.203958069, -4.537291403}
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