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Simplifying 3x2 + x = 111 Reorder the terms: x + 3x2 = 111 Solving x + 3x2 = 111 Solving for variable 'x'. Reorder the terms: -111 + x + 3x2 = 111 + -111 Combine like terms: 111 + -111 = 0 -111 + x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -37 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '37' to each side of the equation. -37 + 0.3333333333x + 37 + x2 = 0 + 37 Reorder the terms: -37 + 37 + 0.3333333333x + x2 = 0 + 37 Combine like terms: -37 + 37 = 0 0 + 0.3333333333x + x2 = 0 + 37 0.3333333333x + x2 = 0 + 37 Combine like terms: 0 + 37 = 37 0.3333333333x + x2 = 37 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333x + 0.25 + x2 = 37 + 0.25 Reorder the terms: 0.25 + 0.3333333333x + x2 = 37 + 0.25 Combine like terms: 37 + 0.25 = 37.25 0.25 + 0.3333333333x + x2 = 37.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 37.25 Calculate the square root of the right side: 6.103277808 Break this problem into two subproblems by setting (x + 0.5) equal to 6.103277808 and -6.103277808.Subproblem 1
x + 0.5 = 6.103277808 Simplifying x + 0.5 = 6.103277808 Reorder the terms: 0.5 + x = 6.103277808 Solving 0.5 + x = 6.103277808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.103277808 + -0.5 x = 6.103277808 + -0.5 Combine like terms: 6.103277808 + -0.5 = 5.603277808 x = 5.603277808 Simplifying x = 5.603277808Subproblem 2
x + 0.5 = -6.103277808 Simplifying x + 0.5 = -6.103277808 Reorder the terms: 0.5 + x = -6.103277808 Solving 0.5 + x = -6.103277808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.103277808 + -0.5 x = -6.103277808 + -0.5 Combine like terms: -6.103277808 + -0.5 = -6.603277808 x = -6.603277808 Simplifying x = -6.603277808Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.603277808, -6.603277808}
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