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Simplifying (6x2 + -5x) + -1(2x2 + 3x + -2) = 0 Reorder the terms: (-5x + 6x2) + -1(2x2 + 3x + -2) = 0 Remove parenthesis around (-5x + 6x2) -5x + 6x2 + -1(2x2 + 3x + -2) = 0 Reorder the terms: -5x + 6x2 + -1(-2 + 3x + 2x2) = 0 -5x + 6x2 + (-2 * -1 + 3x * -1 + 2x2 * -1) = 0 -5x + 6x2 + (2 + -3x + -2x2) = 0 Reorder the terms: 2 + -5x + -3x + 6x2 + -2x2 = 0 Combine like terms: -5x + -3x = -8x 2 + -8x + 6x2 + -2x2 = 0 Combine like terms: 6x2 + -2x2 = 4x2 2 + -8x + 4x2 = 0 Solving 2 + -8x + 4x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -4x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + -4x + 2x2)' equal to zero and attempt to solve: Simplifying 1 + -4x + 2x2 = 0 Solving 1 + -4x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -2x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2x + x2 = 0 + -0.5 -2x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2x + x2 = -0.5 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.5 + 1 Reorder the terms: 1 + -2x + x2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + -2x + x2 = 0.5 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
x + -1 = 0.707106781 Simplifying x + -1 = 0.707106781 Reorder the terms: -1 + x = 0.707106781 Solving -1 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.707106781 + 1 x = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 x = 1.707106781 Simplifying x = 1.707106781Subproblem 2
x + -1 = -0.707106781 Simplifying x + -1 = -0.707106781 Reorder the terms: -1 + x = -0.707106781 Solving -1 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.707106781 + 1 x = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 x = 0.292893219 Simplifying x = 0.292893219Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.707106781, 0.292893219}Solution
x = {1.707106781, 0.292893219}
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