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Simplifying 14x(3x + -5) = 82 Reorder the terms: 14x(-5 + 3x) = 82 (-5 * 14x + 3x * 14x) = 82 (-70x + 42x2) = 82 Solving -70x + 42x2 = 82 Solving for variable 'x'. Reorder the terms: -82 + -70x + 42x2 = 82 + -82 Combine like terms: 82 + -82 = 0 -82 + -70x + 42x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-41 + -35x + 21x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-41 + -35x + 21x2)' equal to zero and attempt to solve: Simplifying -41 + -35x + 21x2 = 0 Solving -41 + -35x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -1.952380952 + -1.666666667x + x2 = 0 Move the constant term to the right: Add '1.952380952' to each side of the equation. -1.952380952 + -1.666666667x + 1.952380952 + x2 = 0 + 1.952380952 Reorder the terms: -1.952380952 + 1.952380952 + -1.666666667x + x2 = 0 + 1.952380952 Combine like terms: -1.952380952 + 1.952380952 = 0.000000000 0.000000000 + -1.666666667x + x2 = 0 + 1.952380952 -1.666666667x + x2 = 0 + 1.952380952 Combine like terms: 0 + 1.952380952 = 1.952380952 -1.666666667x + x2 = 1.952380952 The x term is -1.666666667x. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667x + 0.6944444447 + x2 = 1.952380952 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667x + x2 = 1.952380952 + 0.6944444447 Combine like terms: 1.952380952 + 0.6944444447 = 2.6468253967 0.6944444447 + -1.666666667x + x2 = 2.6468253967 Factor a perfect square on the left side: (x + -0.8333333335)(x + -0.8333333335) = 2.6468253967 Calculate the square root of the right side: 1.626906696 Break this problem into two subproblems by setting (x + -0.8333333335) equal to 1.626906696 and -1.626906696.Subproblem 1
x + -0.8333333335 = 1.626906696 Simplifying x + -0.8333333335 = 1.626906696 Reorder the terms: -0.8333333335 + x = 1.626906696 Solving -0.8333333335 + x = 1.626906696 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 1.626906696 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 1.626906696 + 0.8333333335 x = 1.626906696 + 0.8333333335 Combine like terms: 1.626906696 + 0.8333333335 = 2.4602400295 x = 2.4602400295 Simplifying x = 2.4602400295Subproblem 2
x + -0.8333333335 = -1.626906696 Simplifying x + -0.8333333335 = -1.626906696 Reorder the terms: -0.8333333335 + x = -1.626906696 Solving -0.8333333335 + x = -1.626906696 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -1.626906696 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -1.626906696 + 0.8333333335 x = -1.626906696 + 0.8333333335 Combine like terms: -1.626906696 + 0.8333333335 = -0.7935733625 x = -0.7935733625 Simplifying x = -0.7935733625Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.4602400295, -0.7935733625}Solution
x = {2.4602400295, -0.7935733625}
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