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Simplifying 75x2 + -30x + 3 = 8 Reorder the terms: 3 + -30x + 75x2 = 8 Solving 3 + -30x + 75x2 = 8 Solving for variable 'x'. Reorder the terms: 3 + -8 + -30x + 75x2 = 8 + -8 Combine like terms: 3 + -8 = -5 -5 + -30x + 75x2 = 8 + -8 Combine like terms: 8 + -8 = 0 -5 + -30x + 75x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-1 + -6x + 15x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-1 + -6x + 15x2)' equal to zero and attempt to solve: Simplifying -1 + -6x + 15x2 = 0 Solving -1 + -6x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.06666666667 + -0.4x + x2 = 0 Move the constant term to the right: Add '0.06666666667' to each side of the equation. -0.06666666667 + -0.4x + 0.06666666667 + x2 = 0 + 0.06666666667 Reorder the terms: -0.06666666667 + 0.06666666667 + -0.4x + x2 = 0 + 0.06666666667 Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000 0.00000000000 + -0.4x + x2 = 0 + 0.06666666667 -0.4x + x2 = 0 + 0.06666666667 Combine like terms: 0 + 0.06666666667 = 0.06666666667 -0.4x + x2 = 0.06666666667 The x term is -0.4x. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4x + 0.04 + x2 = 0.06666666667 + 0.04 Reorder the terms: 0.04 + -0.4x + x2 = 0.06666666667 + 0.04 Combine like terms: 0.06666666667 + 0.04 = 0.10666666667 0.04 + -0.4x + x2 = 0.10666666667 Factor a perfect square on the left side: (x + -0.2)(x + -0.2) = 0.10666666667 Calculate the square root of the right side: 0.326598632 Break this problem into two subproblems by setting (x + -0.2) equal to 0.326598632 and -0.326598632.Subproblem 1
x + -0.2 = 0.326598632 Simplifying x + -0.2 = 0.326598632 Reorder the terms: -0.2 + x = 0.326598632 Solving -0.2 + x = 0.326598632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + x = 0.326598632 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + x = 0.326598632 + 0.2 x = 0.326598632 + 0.2 Combine like terms: 0.326598632 + 0.2 = 0.526598632 x = 0.526598632 Simplifying x = 0.526598632Subproblem 2
x + -0.2 = -0.326598632 Simplifying x + -0.2 = -0.326598632 Reorder the terms: -0.2 + x = -0.326598632 Solving -0.2 + x = -0.326598632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + x = -0.326598632 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + x = -0.326598632 + 0.2 x = -0.326598632 + 0.2 Combine like terms: -0.326598632 + 0.2 = -0.126598632 x = -0.126598632 Simplifying x = -0.126598632Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.526598632, -0.126598632}Solution
x = {0.526598632, -0.126598632}
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