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Simplifying 10x + (8x + -16) + (12x + -8) + (7x + 2) + (9x + 4)(6x + 10) = 0 Reorder the terms: 10x + (-16 + 8x) + (12x + -8) + (7x + 2) + (9x + 4)(6x + 10) = 0 Remove parenthesis around (-16 + 8x) 10x + -16 + 8x + (12x + -8) + (7x + 2) + (9x + 4)(6x + 10) = 0 Reorder the terms: 10x + -16 + 8x + (-8 + 12x) + (7x + 2) + (9x + 4)(6x + 10) = 0 Remove parenthesis around (-8 + 12x) 10x + -16 + 8x + -8 + 12x + (7x + 2) + (9x + 4)(6x + 10) = 0 Reorder the terms: 10x + -16 + 8x + -8 + 12x + (2 + 7x) + (9x + 4)(6x + 10) = 0 Remove parenthesis around (2 + 7x) 10x + -16 + 8x + -8 + 12x + 2 + 7x + (9x + 4)(6x + 10) = 0 Reorder the terms: 10x + -16 + 8x + -8 + 12x + 2 + 7x + (4 + 9x)(6x + 10) = 0 Reorder the terms: 10x + -16 + 8x + -8 + 12x + 2 + 7x + (4 + 9x)(10 + 6x) = 0 Multiply (4 + 9x) * (10 + 6x) 10x + -16 + 8x + -8 + 12x + 2 + 7x + (4(10 + 6x) + 9x * (10 + 6x)) = 0 10x + -16 + 8x + -8 + 12x + 2 + 7x + ((10 * 4 + 6x * 4) + 9x * (10 + 6x)) = 0 10x + -16 + 8x + -8 + 12x + 2 + 7x + ((40 + 24x) + 9x * (10 + 6x)) = 0 10x + -16 + 8x + -8 + 12x + 2 + 7x + (40 + 24x + (10 * 9x + 6x * 9x)) = 0 10x + -16 + 8x + -8 + 12x + 2 + 7x + (40 + 24x + (90x + 54x2)) = 0 Combine like terms: 24x + 90x = 114x 10x + -16 + 8x + -8 + 12x + 2 + 7x + (40 + 114x + 54x2) = 0 Reorder the terms: -16 + -8 + 2 + 40 + 10x + 8x + 12x + 7x + 114x + 54x2 = 0 Combine like terms: -16 + -8 = -24 -24 + 2 + 40 + 10x + 8x + 12x + 7x + 114x + 54x2 = 0 Combine like terms: -24 + 2 = -22 -22 + 40 + 10x + 8x + 12x + 7x + 114x + 54x2 = 0 Combine like terms: -22 + 40 = 18 18 + 10x + 8x + 12x + 7x + 114x + 54x2 = 0 Combine like terms: 10x + 8x = 18x 18 + 18x + 12x + 7x + 114x + 54x2 = 0 Combine like terms: 18x + 12x = 30x 18 + 30x + 7x + 114x + 54x2 = 0 Combine like terms: 30x + 7x = 37x 18 + 37x + 114x + 54x2 = 0 Combine like terms: 37x + 114x = 151x 18 + 151x + 54x2 = 0 Solving 18 + 151x + 54x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 54 the coefficient of the squared term: Divide each side by '54'. 0.3333333333 + 2.796296296x + x2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 2.796296296x + -0.3333333333 + x2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 2.796296296x + x2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 2.796296296x + x2 = 0 + -0.3333333333 2.796296296x + x2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 2.796296296x + x2 = -0.3333333333 The x term is 2.796296296x. Take half its coefficient (1.398148148). Square it (1.954818244) and add it to both sides. Add '1.954818244' to each side of the equation. 2.796296296x + 1.954818244 + x2 = -0.3333333333 + 1.954818244 Reorder the terms: 1.954818244 + 2.796296296x + x2 = -0.3333333333 + 1.954818244 Combine like terms: -0.3333333333 + 1.954818244 = 1.6214849107 1.954818244 + 2.796296296x + x2 = 1.6214849107 Factor a perfect square on the left side: (x + 1.398148148)(x + 1.398148148) = 1.6214849107 Calculate the square root of the right side: 1.273375401 Break this problem into two subproblems by setting (x + 1.398148148) equal to 1.273375401 and -1.273375401.Subproblem 1
x + 1.398148148 = 1.273375401 Simplifying x + 1.398148148 = 1.273375401 Reorder the terms: 1.398148148 + x = 1.273375401 Solving 1.398148148 + x = 1.273375401 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.398148148' to each side of the equation. 1.398148148 + -1.398148148 + x = 1.273375401 + -1.398148148 Combine like terms: 1.398148148 + -1.398148148 = 0.000000000 0.000000000 + x = 1.273375401 + -1.398148148 x = 1.273375401 + -1.398148148 Combine like terms: 1.273375401 + -1.398148148 = -0.124772747 x = -0.124772747 Simplifying x = -0.124772747Subproblem 2
x + 1.398148148 = -1.273375401 Simplifying x + 1.398148148 = -1.273375401 Reorder the terms: 1.398148148 + x = -1.273375401 Solving 1.398148148 + x = -1.273375401 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.398148148' to each side of the equation. 1.398148148 + -1.398148148 + x = -1.273375401 + -1.398148148 Combine like terms: 1.398148148 + -1.398148148 = 0.000000000 0.000000000 + x = -1.273375401 + -1.398148148 x = -1.273375401 + -1.398148148 Combine like terms: -1.273375401 + -1.398148148 = -2.671523549 x = -2.671523549 Simplifying x = -2.671523549Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.124772747, -2.671523549}
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