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Simplifying 6x(x + -2) = 19x + 24 Reorder the terms: 6x(-2 + x) = 19x + 24 (-2 * 6x + x * 6x) = 19x + 24 (-12x + 6x2) = 19x + 24 Reorder the terms: -12x + 6x2 = 24 + 19x Solving -12x + 6x2 = 24 + 19x Solving for variable 'x'. Reorder the terms: -24 + -12x + -19x + 6x2 = 24 + 19x + -24 + -19x Combine like terms: -12x + -19x = -31x -24 + -31x + 6x2 = 24 + 19x + -24 + -19x Reorder the terms: -24 + -31x + 6x2 = 24 + -24 + 19x + -19x Combine like terms: 24 + -24 = 0 -24 + -31x + 6x2 = 0 + 19x + -19x -24 + -31x + 6x2 = 19x + -19x Combine like terms: 19x + -19x = 0 -24 + -31x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4 + -5.166666667x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -5.166666667x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + -5.166666667x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -5.166666667x + x2 = 0 + 4 -5.166666667x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 -5.166666667x + x2 = 4 The x term is -5.166666667x. Take half its coefficient (-2.583333334). Square it (6.673611115) and add it to both sides. Add '6.673611115' to each side of the equation. -5.166666667x + 6.673611115 + x2 = 4 + 6.673611115 Reorder the terms: 6.673611115 + -5.166666667x + x2 = 4 + 6.673611115 Combine like terms: 4 + 6.673611115 = 10.673611115 6.673611115 + -5.166666667x + x2 = 10.673611115 Factor a perfect square on the left side: (x + -2.583333334)(x + -2.583333334) = 10.673611115 Calculate the square root of the right side: 3.267049298 Break this problem into two subproblems by setting (x + -2.583333334) equal to 3.267049298 and -3.267049298.Subproblem 1
x + -2.583333334 = 3.267049298 Simplifying x + -2.583333334 = 3.267049298 Reorder the terms: -2.583333334 + x = 3.267049298 Solving -2.583333334 + x = 3.267049298 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.583333334' to each side of the equation. -2.583333334 + 2.583333334 + x = 3.267049298 + 2.583333334 Combine like terms: -2.583333334 + 2.583333334 = 0.000000000 0.000000000 + x = 3.267049298 + 2.583333334 x = 3.267049298 + 2.583333334 Combine like terms: 3.267049298 + 2.583333334 = 5.850382632 x = 5.850382632 Simplifying x = 5.850382632Subproblem 2
x + -2.583333334 = -3.267049298 Simplifying x + -2.583333334 = -3.267049298 Reorder the terms: -2.583333334 + x = -3.267049298 Solving -2.583333334 + x = -3.267049298 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.583333334' to each side of the equation. -2.583333334 + 2.583333334 + x = -3.267049298 + 2.583333334 Combine like terms: -2.583333334 + 2.583333334 = 0.000000000 0.000000000 + x = -3.267049298 + 2.583333334 x = -3.267049298 + 2.583333334 Combine like terms: -3.267049298 + 2.583333334 = -0.683715964 x = -0.683715964 Simplifying x = -0.683715964Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.850382632, -0.683715964}
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