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Simplifying -13x(x + -4) + 8(2x + -3) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) Reorder the terms: -13x(-4 + x) + 8(2x + -3) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) (-4 * -13x + x * -13x) + 8(2x + -3) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) (52x + -13x2) + 8(2x + -3) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) Reorder the terms: 52x + -13x2 + 8(-3 + 2x) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) 52x + -13x2 + (-3 * 8 + 2x * 8) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) 52x + -13x2 + (-24 + 16x) = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) Reorder the terms: -24 + 52x + 16x + -13x2 = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) Combine like terms: 52x + 16x = 68x -24 + 68x + -13x2 = -1(x + 5) + (x + -5) + (x + -5) + -10(x + 1) Reorder the terms: -24 + 68x + -13x2 = -1(5 + x) + (x + -5) + (x + -5) + -10(x + 1) -24 + 68x + -13x2 = (5 * -1 + x * -1) + (x + -5) + (x + -5) + -10(x + 1) -24 + 68x + -13x2 = (-5 + -1x) + (x + -5) + (x + -5) + -10(x + 1) Reorder the terms: -24 + 68x + -13x2 = -5 + -1x + (-5 + x) + (x + -5) + -10(x + 1) Remove parenthesis around (-5 + x) -24 + 68x + -13x2 = -5 + -1x + -5 + x + (x + -5) + -10(x + 1) Reorder the terms: -24 + 68x + -13x2 = -5 + -1x + -5 + x + (-5 + x) + -10(x + 1) Remove parenthesis around (-5 + x) -24 + 68x + -13x2 = -5 + -1x + -5 + x + -5 + x + -10(x + 1) Reorder the terms: -24 + 68x + -13x2 = -5 + -1x + -5 + x + -5 + x + -10(1 + x) -24 + 68x + -13x2 = -5 + -1x + -5 + x + -5 + x + (1 * -10 + x * -10) -24 + 68x + -13x2 = -5 + -1x + -5 + x + -5 + x + (-10 + -10x) Reorder the terms: -24 + 68x + -13x2 = -5 + -5 + -5 + -10 + -1x + x + x + -10x Combine like terms: -5 + -5 = -10 -24 + 68x + -13x2 = -10 + -5 + -10 + -1x + x + x + -10x Combine like terms: -10 + -5 = -15 -24 + 68x + -13x2 = -15 + -10 + -1x + x + x + -10x Combine like terms: -15 + -10 = -25 -24 + 68x + -13x2 = -25 + -1x + x + x + -10x Combine like terms: -1x + x = 0 -24 + 68x + -13x2 = -25 + 0 + x + -10x -24 + 68x + -13x2 = -25 + x + -10x Combine like terms: x + -10x = -9x -24 + 68x + -13x2 = -25 + -9x Solving -24 + 68x + -13x2 = -25 + -9x Solving for variable 'x'. Reorder the terms: -24 + 25 + 68x + 9x + -13x2 = -25 + -9x + 25 + 9x Combine like terms: -24 + 25 = 1 1 + 68x + 9x + -13x2 = -25 + -9x + 25 + 9x Combine like terms: 68x + 9x = 77x 1 + 77x + -13x2 = -25 + -9x + 25 + 9x Reorder the terms: 1 + 77x + -13x2 = -25 + 25 + -9x + 9x Combine like terms: -25 + 25 = 0 1 + 77x + -13x2 = 0 + -9x + 9x 1 + 77x + -13x2 = -9x + 9x Combine like terms: -9x + 9x = 0 1 + 77x + -13x2 = 0 Begin completing the square. Divide all terms by -13 the coefficient of the squared term: Divide each side by '-13'. -0.07692307692 + -5.923076923x + x2 = 0 Move the constant term to the right: Add '0.07692307692' to each side of the equation. -0.07692307692 + -5.923076923x + 0.07692307692 + x2 = 0 + 0.07692307692 Reorder the terms: -0.07692307692 + 0.07692307692 + -5.923076923x + x2 = 0 + 0.07692307692 Combine like terms: -0.07692307692 + 0.07692307692 = 0.00000000000 0.00000000000 + -5.923076923x + x2 = 0 + 0.07692307692 -5.923076923x + x2 = 0 + 0.07692307692 Combine like terms: 0 + 0.07692307692 = 0.07692307692 -5.923076923x + x2 = 0.07692307692 The x term is -5.923076923x. Take half its coefficient (-2.961538462). Square it (8.770710062) and add it to both sides. Add '8.770710062' to each side of the equation. -5.923076923x + 8.770710062 + x2 = 0.07692307692 + 8.770710062 Reorder the terms: 8.770710062 + -5.923076923x + x2 = 0.07692307692 + 8.770710062 Combine like terms: 0.07692307692 + 8.770710062 = 8.84763313892 8.770710062 + -5.923076923x + x2 = 8.84763313892 Factor a perfect square on the left side: (x + -2.961538462)(x + -2.961538462) = 8.84763313892 Calculate the square root of the right side: 2.974497124 Break this problem into two subproblems by setting (x + -2.961538462) equal to 2.974497124 and -2.974497124.Subproblem 1
x + -2.961538462 = 2.974497124 Simplifying x + -2.961538462 = 2.974497124 Reorder the terms: -2.961538462 + x = 2.974497124 Solving -2.961538462 + x = 2.974497124 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.961538462' to each side of the equation. -2.961538462 + 2.961538462 + x = 2.974497124 + 2.961538462 Combine like terms: -2.961538462 + 2.961538462 = 0.000000000 0.000000000 + x = 2.974497124 + 2.961538462 x = 2.974497124 + 2.961538462 Combine like terms: 2.974497124 + 2.961538462 = 5.936035586 x = 5.936035586 Simplifying x = 5.936035586Subproblem 2
x + -2.961538462 = -2.974497124 Simplifying x + -2.961538462 = -2.974497124 Reorder the terms: -2.961538462 + x = -2.974497124 Solving -2.961538462 + x = -2.974497124 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.961538462' to each side of the equation. -2.961538462 + 2.961538462 + x = -2.974497124 + 2.961538462 Combine like terms: -2.961538462 + 2.961538462 = 0.000000000 0.000000000 + x = -2.974497124 + 2.961538462 x = -2.974497124 + 2.961538462 Combine like terms: -2.974497124 + 2.961538462 = -0.012958662 x = -0.012958662 Simplifying x = -0.012958662Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.936035586, -0.012958662}
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