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Simplifying 32x + 12x2 = 10 Solving 32x + 12x2 = 10 Solving for variable 'x'. Reorder the terms: -10 + 32x + 12x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 32x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + 16x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + 16x + 6x2)' equal to zero and attempt to solve: Simplifying -5 + 16x + 6x2 = 0 Solving -5 + 16x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 2.666666667x + 0.8333333333 + x2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 2.666666667x + x2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 2.666666667x + x2 = 0 + 0.8333333333 2.666666667x + x2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 2.666666667x + x2 = 0.8333333333 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 0.8333333333 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 0.8333333333 + 1.777777780 Combine like terms: 0.8333333333 + 1.777777780 = 2.6111111133 1.777777780 + 2.666666667x + x2 = 2.6111111133 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 2.6111111133 Calculate the square root of the right side: 1.615893286 Break this problem into two subproblems by setting (x + 1.333333334) equal to 1.615893286 and -1.615893286.Subproblem 1
x + 1.333333334 = 1.615893286 Simplifying x + 1.333333334 = 1.615893286 Reorder the terms: 1.333333334 + x = 1.615893286 Solving 1.333333334 + x = 1.615893286 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 1.615893286 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.615893286 + -1.333333334 x = 1.615893286 + -1.333333334 Combine like terms: 1.615893286 + -1.333333334 = 0.282559952 x = 0.282559952 Simplifying x = 0.282559952Subproblem 2
x + 1.333333334 = -1.615893286 Simplifying x + 1.333333334 = -1.615893286 Reorder the terms: 1.333333334 + x = -1.615893286 Solving 1.333333334 + x = -1.615893286 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -1.615893286 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.615893286 + -1.333333334 x = -1.615893286 + -1.333333334 Combine like terms: -1.615893286 + -1.333333334 = -2.94922662 x = -2.94922662 Simplifying x = -2.94922662Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.282559952, -2.94922662}Solution
x = {0.282559952, -2.94922662}
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