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Simplifying (2x + -1)(3x + 4) = 180 Reorder the terms: (-1 + 2x)(3x + 4) = 180 Reorder the terms: (-1 + 2x)(4 + 3x) = 180 Multiply (-1 + 2x) * (4 + 3x) (-1(4 + 3x) + 2x * (4 + 3x)) = 180 ((4 * -1 + 3x * -1) + 2x * (4 + 3x)) = 180 ((-4 + -3x) + 2x * (4 + 3x)) = 180 (-4 + -3x + (4 * 2x + 3x * 2x)) = 180 (-4 + -3x + (8x + 6x2)) = 180 Combine like terms: -3x + 8x = 5x (-4 + 5x + 6x2) = 180 Solving -4 + 5x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: -4 + -180 + 5x + 6x2 = 180 + -180 Combine like terms: -4 + -180 = -184 -184 + 5x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -184 + 5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -30.66666667 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '30.66666667' to each side of the equation. -30.66666667 + 0.8333333333x + 30.66666667 + x2 = 0 + 30.66666667 Reorder the terms: -30.66666667 + 30.66666667 + 0.8333333333x + x2 = 0 + 30.66666667 Combine like terms: -30.66666667 + 30.66666667 = 0.00000000 0.00000000 + 0.8333333333x + x2 = 0 + 30.66666667 0.8333333333x + x2 = 0 + 30.66666667 Combine like terms: 0 + 30.66666667 = 30.66666667 0.8333333333x + x2 = 30.66666667 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 = 30.66666667 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 30.66666667 + 0.1736111111 Combine like terms: 30.66666667 + 0.1736111111 = 30.8402777811 0.1736111111 + 0.8333333333x + x2 = 30.8402777811 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 30.8402777811 Calculate the square root of the right side: 5.553402361 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 5.553402361 and -5.553402361.Subproblem 1
x + 0.4166666667 = 5.553402361 Simplifying x + 0.4166666667 = 5.553402361 Reorder the terms: 0.4166666667 + x = 5.553402361 Solving 0.4166666667 + x = 5.553402361 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 = 5.553402361 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 5.553402361 + -0.4166666667 x = 5.553402361 + -0.4166666667 Combine like terms: 5.553402361 + -0.4166666667 = 5.1367356943 x = 5.1367356943 Simplifying x = 5.1367356943Subproblem 2
x + 0.4166666667 = -5.553402361 Simplifying x + 0.4166666667 = -5.553402361 Reorder the terms: 0.4166666667 + x = -5.553402361 Solving 0.4166666667 + x = -5.553402361 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 = -5.553402361 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -5.553402361 + -0.4166666667 x = -5.553402361 + -0.4166666667 Combine like terms: -5.553402361 + -0.4166666667 = -5.9700690277 x = -5.9700690277 Simplifying x = -5.9700690277Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.1367356943, -5.9700690277}
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