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Simplifying (6x + -8)(2x + 7) + -2.5(3 + -1x) = 0 Reorder the terms: (-8 + 6x)(2x + 7) + -2.5(3 + -1x) = 0 Reorder the terms: (-8 + 6x)(7 + 2x) + -2.5(3 + -1x) = 0 Multiply (-8 + 6x) * (7 + 2x) (-8(7 + 2x) + 6x * (7 + 2x)) + -2.5(3 + -1x) = 0 ((7 * -8 + 2x * -8) + 6x * (7 + 2x)) + -2.5(3 + -1x) = 0 ((-56 + -16x) + 6x * (7 + 2x)) + -2.5(3 + -1x) = 0 (-56 + -16x + (7 * 6x + 2x * 6x)) + -2.5(3 + -1x) = 0 (-56 + -16x + (42x + 12x2)) + -2.5(3 + -1x) = 0 Combine like terms: -16x + 42x = 26x (-56 + 26x + 12x2) + -2.5(3 + -1x) = 0 -56 + 26x + 12x2 + (3 * -2.5 + -1x * -2.5) = 0 -56 + 26x + 12x2 + (-7.5 + 2.5x) = 0 Reorder the terms: -56 + -7.5 + 26x + 2.5x + 12x2 = 0 Combine like terms: -56 + -7.5 = -63.5 -63.5 + 26x + 2.5x + 12x2 = 0 Combine like terms: 26x + 2.5x = 28.5x -63.5 + 28.5x + 12x2 = 0 Solving -63.5 + 28.5x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -5.291666667 + 2.375x + x2 = 0 Move the constant term to the right: Add '5.291666667' to each side of the equation. -5.291666667 + 2.375x + 5.291666667 + x2 = 0 + 5.291666667 Reorder the terms: -5.291666667 + 5.291666667 + 2.375x + x2 = 0 + 5.291666667 Combine like terms: -5.291666667 + 5.291666667 = 0.000000000 0.000000000 + 2.375x + x2 = 0 + 5.291666667 2.375x + x2 = 0 + 5.291666667 Combine like terms: 0 + 5.291666667 = 5.291666667 2.375x + x2 = 5.291666667 The x term is 2.375x. Take half its coefficient (1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. 2.375x + 1.41015625 + x2 = 5.291666667 + 1.41015625 Reorder the terms: 1.41015625 + 2.375x + x2 = 5.291666667 + 1.41015625 Combine like terms: 5.291666667 + 1.41015625 = 6.701822917 1.41015625 + 2.375x + x2 = 6.701822917 Factor a perfect square on the left side: (x + 1.1875)(x + 1.1875) = 6.701822917 Calculate the square root of the right side: 2.588787924 Break this problem into two subproblems by setting (x + 1.1875) equal to 2.588787924 and -2.588787924.Subproblem 1
x + 1.1875 = 2.588787924 Simplifying x + 1.1875 = 2.588787924 Reorder the terms: 1.1875 + x = 2.588787924 Solving 1.1875 + x = 2.588787924 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.1875' to each side of the equation. 1.1875 + -1.1875 + x = 2.588787924 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + x = 2.588787924 + -1.1875 x = 2.588787924 + -1.1875 Combine like terms: 2.588787924 + -1.1875 = 1.401287924 x = 1.401287924 Simplifying x = 1.401287924Subproblem 2
x + 1.1875 = -2.588787924 Simplifying x + 1.1875 = -2.588787924 Reorder the terms: 1.1875 + x = -2.588787924 Solving 1.1875 + x = -2.588787924 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.1875' to each side of the equation. 1.1875 + -1.1875 + x = -2.588787924 + -1.1875 Combine like terms: 1.1875 + -1.1875 = 0.0000 0.0000 + x = -2.588787924 + -1.1875 x = -2.588787924 + -1.1875 Combine like terms: -2.588787924 + -1.1875 = -3.776287924 x = -3.776287924 Simplifying x = -3.776287924Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.401287924, -3.776287924}
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