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Simplifying 30x2 + -10x = 5x + -1 Reorder the terms: -10x + 30x2 = 5x + -1 Reorder the terms: -10x + 30x2 = -1 + 5x Solving -10x + 30x2 = -1 + 5x Solving for variable 'x'. Reorder the terms: 1 + -10x + -5x + 30x2 = -1 + 5x + 1 + -5x Combine like terms: -10x + -5x = -15x 1 + -15x + 30x2 = -1 + 5x + 1 + -5x Reorder the terms: 1 + -15x + 30x2 = -1 + 1 + 5x + -5x Combine like terms: -1 + 1 = 0 1 + -15x + 30x2 = 0 + 5x + -5x 1 + -15x + 30x2 = 5x + -5x Combine like terms: 5x + -5x = 0 1 + -15x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. 0.03333333333 + -0.5x + x2 = 0 Move the constant term to the right: Add '-0.03333333333' to each side of the equation. 0.03333333333 + -0.5x + -0.03333333333 + x2 = 0 + -0.03333333333 Reorder the terms: 0.03333333333 + -0.03333333333 + -0.5x + x2 = 0 + -0.03333333333 Combine like terms: 0.03333333333 + -0.03333333333 = 0.00000000000 0.00000000000 + -0.5x + x2 = 0 + -0.03333333333 -0.5x + x2 = 0 + -0.03333333333 Combine like terms: 0 + -0.03333333333 = -0.03333333333 -0.5x + x2 = -0.03333333333 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = -0.03333333333 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = -0.03333333333 + 0.0625 Combine like terms: -0.03333333333 + 0.0625 = 0.02916666667 0.0625 + -0.5x + x2 = 0.02916666667 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 0.02916666667 Calculate the square root of the right side: 0.170782513 Break this problem into two subproblems by setting (x + -0.25) equal to 0.170782513 and -0.170782513.Subproblem 1
x + -0.25 = 0.170782513 Simplifying x + -0.25 = 0.170782513 Reorder the terms: -0.25 + x = 0.170782513 Solving -0.25 + x = 0.170782513 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 0.170782513 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 0.170782513 + 0.25 x = 0.170782513 + 0.25 Combine like terms: 0.170782513 + 0.25 = 0.420782513 x = 0.420782513 Simplifying x = 0.420782513Subproblem 2
x + -0.25 = -0.170782513 Simplifying x + -0.25 = -0.170782513 Reorder the terms: -0.25 + x = -0.170782513 Solving -0.25 + x = -0.170782513 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -0.170782513 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -0.170782513 + 0.25 x = -0.170782513 + 0.25 Combine like terms: -0.170782513 + 0.25 = 0.079217487 x = 0.079217487 Simplifying x = 0.079217487Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.420782513, 0.079217487}
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