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Simplifying 12x2 + 41x + -42 = 0 Reorder the terms: -42 + 41x + 12x2 = 0 Solving -42 + 41x + 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'. -3.5 + 3.416666667x + x2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 3.416666667x + 3.5 + x2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 3.416666667x + x2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 3.416666667x + x2 = 0 + 3.5 3.416666667x + x2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 3.416666667x + x2 = 3.5 The x term is 3.416666667x. Take half its coefficient (1.708333334). Square it (2.918402780) and add it to both sides. Add '2.918402780' to each side of the equation. 3.416666667x + 2.918402780 + x2 = 3.5 + 2.918402780 Reorder the terms: 2.918402780 + 3.416666667x + x2 = 3.5 + 2.918402780 Combine like terms: 3.5 + 2.918402780 = 6.41840278 2.918402780 + 3.416666667x + x2 = 6.41840278 Factor a perfect square on the left side: (x + 1.708333334)(x + 1.708333334) = 6.41840278 Calculate the square root of the right side: 2.533456686 Break this problem into two subproblems by setting (x + 1.708333334) equal to 2.533456686 and -2.533456686.Subproblem 1
x + 1.708333334 = 2.533456686 Simplifying x + 1.708333334 = 2.533456686 Reorder the terms: 1.708333334 + x = 2.533456686 Solving 1.708333334 + x = 2.533456686 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.708333334' to each side of the equation. 1.708333334 + -1.708333334 + x = 2.533456686 + -1.708333334 Combine like terms: 1.708333334 + -1.708333334 = 0.000000000 0.000000000 + x = 2.533456686 + -1.708333334 x = 2.533456686 + -1.708333334 Combine like terms: 2.533456686 + -1.708333334 = 0.825123352 x = 0.825123352 Simplifying x = 0.825123352Subproblem 2
x + 1.708333334 = -2.533456686 Simplifying x + 1.708333334 = -2.533456686 Reorder the terms: 1.708333334 + x = -2.533456686 Solving 1.708333334 + x = -2.533456686 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.708333334' to each side of the equation. 1.708333334 + -1.708333334 + x = -2.533456686 + -1.708333334 Combine like terms: 1.708333334 + -1.708333334 = 0.000000000 0.000000000 + x = -2.533456686 + -1.708333334 x = -2.533456686 + -1.708333334 Combine like terms: -2.533456686 + -1.708333334 = -4.24179002 x = -4.24179002 Simplifying x = -4.24179002Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.825123352, -4.24179002}
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