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Simplifying (2x + -7)(4x + -1) + -1(5x + -1)(2x + -1) = 0 Reorder the terms: (-7 + 2x)(4x + -1) + -1(5x + -1)(2x + -1) = 0 Reorder the terms: (-7 + 2x)(-1 + 4x) + -1(5x + -1)(2x + -1) = 0 Multiply (-7 + 2x) * (-1 + 4x) (-7(-1 + 4x) + 2x * (-1 + 4x)) + -1(5x + -1)(2x + -1) = 0 ((-1 * -7 + 4x * -7) + 2x * (-1 + 4x)) + -1(5x + -1)(2x + -1) = 0 ((7 + -28x) + 2x * (-1 + 4x)) + -1(5x + -1)(2x + -1) = 0 (7 + -28x + (-1 * 2x + 4x * 2x)) + -1(5x + -1)(2x + -1) = 0 (7 + -28x + (-2x + 8x2)) + -1(5x + -1)(2x + -1) = 0 Combine like terms: -28x + -2x = -30x (7 + -30x + 8x2) + -1(5x + -1)(2x + -1) = 0 Reorder the terms: 7 + -30x + 8x2 + -1(-1 + 5x)(2x + -1) = 0 Reorder the terms: 7 + -30x + 8x2 + -1(-1 + 5x)(-1 + 2x) = 0 Multiply (-1 + 5x) * (-1 + 2x) 7 + -30x + 8x2 + -1(-1(-1 + 2x) + 5x * (-1 + 2x)) = 0 7 + -30x + 8x2 + -1((-1 * -1 + 2x * -1) + 5x * (-1 + 2x)) = 0 7 + -30x + 8x2 + -1((1 + -2x) + 5x * (-1 + 2x)) = 0 7 + -30x + 8x2 + -1(1 + -2x + (-1 * 5x + 2x * 5x)) = 0 7 + -30x + 8x2 + -1(1 + -2x + (-5x + 10x2)) = 0 Combine like terms: -2x + -5x = -7x 7 + -30x + 8x2 + -1(1 + -7x + 10x2) = 0 7 + -30x + 8x2 + (1 * -1 + -7x * -1 + 10x2 * -1) = 0 7 + -30x + 8x2 + (-1 + 7x + -10x2) = 0 Reorder the terms: 7 + -1 + -30x + 7x + 8x2 + -10x2 = 0 Combine like terms: 7 + -1 = 6 6 + -30x + 7x + 8x2 + -10x2 = 0 Combine like terms: -30x + 7x = -23x 6 + -23x + 8x2 + -10x2 = 0 Combine like terms: 8x2 + -10x2 = -2x2 6 + -23x + -2x2 = 0 Solving 6 + -23x + -2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3 + 11.5x + x2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 11.5x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + 11.5x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 11.5x + x2 = 0 + 3 11.5x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 11.5x + x2 = 3 The x term is 11.5x. Take half its coefficient (5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. 11.5x + 33.0625 + x2 = 3 + 33.0625 Reorder the terms: 33.0625 + 11.5x + x2 = 3 + 33.0625 Combine like terms: 3 + 33.0625 = 36.0625 33.0625 + 11.5x + x2 = 36.0625 Factor a perfect square on the left side: (x + 5.75)(x + 5.75) = 36.0625 Calculate the square root of the right side: 6.005206075 Break this problem into two subproblems by setting (x + 5.75) equal to 6.005206075 and -6.005206075.Subproblem 1
x + 5.75 = 6.005206075 Simplifying x + 5.75 = 6.005206075 Reorder the terms: 5.75 + x = 6.005206075 Solving 5.75 + x = 6.005206075 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + x = 6.005206075 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + x = 6.005206075 + -5.75 x = 6.005206075 + -5.75 Combine like terms: 6.005206075 + -5.75 = 0.255206075 x = 0.255206075 Simplifying x = 0.255206075Subproblem 2
x + 5.75 = -6.005206075 Simplifying x + 5.75 = -6.005206075 Reorder the terms: 5.75 + x = -6.005206075 Solving 5.75 + x = -6.005206075 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + x = -6.005206075 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + x = -6.005206075 + -5.75 x = -6.005206075 + -5.75 Combine like terms: -6.005206075 + -5.75 = -11.755206075 x = -11.755206075 Simplifying x = -11.755206075Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.255206075, -11.755206075}
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