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Simplifying 3x(x + 1) + 4(x + 2) = 32 Reorder the terms: 3x(1 + x) + 4(x + 2) = 32 (1 * 3x + x * 3x) + 4(x + 2) = 32 (3x + 3x2) + 4(x + 2) = 32 Reorder the terms: 3x + 3x2 + 4(2 + x) = 32 3x + 3x2 + (2 * 4 + x * 4) = 32 3x + 3x2 + (8 + 4x) = 32 Reorder the terms: 8 + 3x + 4x + 3x2 = 32 Combine like terms: 3x + 4x = 7x 8 + 7x + 3x2 = 32 Solving 8 + 7x + 3x2 = 32 Solving for variable 'x'. Reorder the terms: 8 + -32 + 7x + 3x2 = 32 + -32 Combine like terms: 8 + -32 = -24 -24 + 7x + 3x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -24 + 7x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -8 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 2.333333333x + 8 + x2 = 0 + 8 Reorder the terms: -8 + 8 + 2.333333333x + x2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 2.333333333x + x2 = 0 + 8 2.333333333x + x2 = 0 + 8 Combine like terms: 0 + 8 = 8 2.333333333x + x2 = 8 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 = 8 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 8 + 1.361111112 Combine like terms: 8 + 1.361111112 = 9.361111112 1.361111112 + 2.333333333x + x2 = 9.361111112 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 9.361111112 Calculate the square root of the right side: 3.059593292 Break this problem into two subproblems by setting (x + 1.166666667) equal to 3.059593292 and -3.059593292.Subproblem 1
x + 1.166666667 = 3.059593292 Simplifying x + 1.166666667 = 3.059593292 Reorder the terms: 1.166666667 + x = 3.059593292 Solving 1.166666667 + x = 3.059593292 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 = 3.059593292 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 3.059593292 + -1.166666667 x = 3.059593292 + -1.166666667 Combine like terms: 3.059593292 + -1.166666667 = 1.892926625 x = 1.892926625 Simplifying x = 1.892926625Subproblem 2
x + 1.166666667 = -3.059593292 Simplifying x + 1.166666667 = -3.059593292 Reorder the terms: 1.166666667 + x = -3.059593292 Solving 1.166666667 + x = -3.059593292 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 = -3.059593292 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -3.059593292 + -1.166666667 x = -3.059593292 + -1.166666667 Combine like terms: -3.059593292 + -1.166666667 = -4.226259959 x = -4.226259959 Simplifying x = -4.226259959Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.892926625, -4.226259959}
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