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Simplifying (-12 + -24x + 15x2) + -1(-1 + 22x + 3x2) = 0 Remove parenthesis around (-12 + -24x + 15x2) -12 + -24x + 15x2 + -1(-1 + 22x + 3x2) = 0 -12 + -24x + 15x2 + (-1 * -1 + 22x * -1 + 3x2 * -1) = 0 -12 + -24x + 15x2 + (1 + -22x + -3x2) = 0 Reorder the terms: -12 + 1 + -24x + -22x + 15x2 + -3x2 = 0 Combine like terms: -12 + 1 = -11 -11 + -24x + -22x + 15x2 + -3x2 = 0 Combine like terms: -24x + -22x = -46x -11 + -46x + 15x2 + -3x2 = 0 Combine like terms: 15x2 + -3x2 = 12x2 -11 + -46x + 12x2 = 0 Solving -11 + -46x + 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'. -0.9166666667 + -3.833333333x + x2 = 0 Move the constant term to the right: Add '0.9166666667' to each side of the equation. -0.9166666667 + -3.833333333x + 0.9166666667 + x2 = 0 + 0.9166666667 Reorder the terms: -0.9166666667 + 0.9166666667 + -3.833333333x + x2 = 0 + 0.9166666667 Combine like terms: -0.9166666667 + 0.9166666667 = 0.0000000000 0.0000000000 + -3.833333333x + x2 = 0 + 0.9166666667 -3.833333333x + x2 = 0 + 0.9166666667 Combine like terms: 0 + 0.9166666667 = 0.9166666667 -3.833333333x + x2 = 0.9166666667 The x term is -3.833333333x. Take half its coefficient (-1.916666667). Square it (3.673611112) and add it to both sides. Add '3.673611112' to each side of the equation. -3.833333333x + 3.673611112 + x2 = 0.9166666667 + 3.673611112 Reorder the terms: 3.673611112 + -3.833333333x + x2 = 0.9166666667 + 3.673611112 Combine like terms: 0.9166666667 + 3.673611112 = 4.5902777787 3.673611112 + -3.833333333x + x2 = 4.5902777787 Factor a perfect square on the left side: (x + -1.916666667)(x + -1.916666667) = 4.5902777787 Calculate the square root of the right side: 2.142493356 Break this problem into two subproblems by setting (x + -1.916666667) equal to 2.142493356 and -2.142493356.Subproblem 1
x + -1.916666667 = 2.142493356 Simplifying x + -1.916666667 = 2.142493356 Reorder the terms: -1.916666667 + x = 2.142493356 Solving -1.916666667 + x = 2.142493356 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.916666667' to each side of the equation. -1.916666667 + 1.916666667 + x = 2.142493356 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + x = 2.142493356 + 1.916666667 x = 2.142493356 + 1.916666667 Combine like terms: 2.142493356 + 1.916666667 = 4.059160023 x = 4.059160023 Simplifying x = 4.059160023Subproblem 2
x + -1.916666667 = -2.142493356 Simplifying x + -1.916666667 = -2.142493356 Reorder the terms: -1.916666667 + x = -2.142493356 Solving -1.916666667 + x = -2.142493356 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.916666667' to each side of the equation. -1.916666667 + 1.916666667 + x = -2.142493356 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + x = -2.142493356 + 1.916666667 x = -2.142493356 + 1.916666667 Combine like terms: -2.142493356 + 1.916666667 = -0.225826689 x = -0.225826689 Simplifying x = -0.225826689Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.059160023, -0.225826689}
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