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Simplifying (3x + 4)(2x + -1) = 11 Reorder the terms: (4 + 3x)(2x + -1) = 11 Reorder the terms: (4 + 3x)(-1 + 2x) = 11 Multiply (4 + 3x) * (-1 + 2x) (4(-1 + 2x) + 3x * (-1 + 2x)) = 11 ((-1 * 4 + 2x * 4) + 3x * (-1 + 2x)) = 11 ((-4 + 8x) + 3x * (-1 + 2x)) = 11 (-4 + 8x + (-1 * 3x + 2x * 3x)) = 11 (-4 + 8x + (-3x + 6x2)) = 11 Combine like terms: 8x + -3x = 5x (-4 + 5x + 6x2) = 11 Solving -4 + 5x + 6x2 = 11 Solving for variable 'x'. Reorder the terms: -4 + -11 + 5x + 6x2 = 11 + -11 Combine like terms: -4 + -11 = -15 -15 + 5x + 6x2 = 11 + -11 Combine like terms: 11 + -11 = 0 -15 + 5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.5 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.8333333333x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.8333333333x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.8333333333x + x2 = 0 + 2.5 0.8333333333x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 0.8333333333x + x2 = 2.5 The x term is 0.8333333333x. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333x + 0.1736111111 + x2 = 2.5 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 2.5 + 0.1736111111 Combine like terms: 2.5 + 0.1736111111 = 2.6736111111 0.1736111111 + 0.8333333333x + x2 = 2.6736111111 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 2.6736111111 Calculate the square root of the right side: 1.635118073 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 1.635118073 and -1.635118073.Subproblem 1
x + 0.4166666667 = 1.635118073 Simplifying x + 0.4166666667 = 1.635118073 Reorder the terms: 0.4166666667 + x = 1.635118073 Solving 0.4166666667 + x = 1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 1.635118073 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 1.635118073 + -0.4166666667 x = 1.635118073 + -0.4166666667 Combine like terms: 1.635118073 + -0.4166666667 = 1.2184514063 x = 1.2184514063 Simplifying x = 1.2184514063Subproblem 2
x + 0.4166666667 = -1.635118073 Simplifying x + 0.4166666667 = -1.635118073 Reorder the terms: 0.4166666667 + x = -1.635118073 Solving 0.4166666667 + x = -1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -1.635118073 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -1.635118073 + -0.4166666667 x = -1.635118073 + -0.4166666667 Combine like terms: -1.635118073 + -0.4166666667 = -2.0517847397 x = -2.0517847397 Simplifying x = -2.0517847397Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2184514063, -2.0517847397}
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