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Simplifying 3x2 + x + 10 = 180 Reorder the terms: 10 + x + 3x2 = 180 Solving 10 + x + 3x2 = 180 Solving for variable 'x'. Reorder the terms: 10 + -180 + x + 3x2 = 180 + -180 Combine like terms: 10 + -180 = -170 -170 + x + 3x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -170 + x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -56.66666667 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '56.66666667' to each side of the equation. -56.66666667 + 0.3333333333x + 56.66666667 + x2 = 0 + 56.66666667 Reorder the terms: -56.66666667 + 56.66666667 + 0.3333333333x + x2 = 0 + 56.66666667 Combine like terms: -56.66666667 + 56.66666667 = 0.00000000 0.00000000 + 0.3333333333x + x2 = 0 + 56.66666667 0.3333333333x + x2 = 0 + 56.66666667 Combine like terms: 0 + 56.66666667 = 56.66666667 0.3333333333x + x2 = 56.66666667 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 = 56.66666667 + 0.25 Reorder the terms: 0.25 + 0.3333333333x + x2 = 56.66666667 + 0.25 Combine like terms: 56.66666667 + 0.25 = 56.91666667 0.25 + 0.3333333333x + x2 = 56.91666667 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 56.91666667 Calculate the square root of the right side: 7.544313532 Break this problem into two subproblems by setting (x + 0.5) equal to 7.544313532 and -7.544313532.Subproblem 1
x + 0.5 = 7.544313532 Simplifying x + 0.5 = 7.544313532 Reorder the terms: 0.5 + x = 7.544313532 Solving 0.5 + x = 7.544313532 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 = 7.544313532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.544313532 + -0.5 x = 7.544313532 + -0.5 Combine like terms: 7.544313532 + -0.5 = 7.044313532 x = 7.044313532 Simplifying x = 7.044313532Subproblem 2
x + 0.5 = -7.544313532 Simplifying x + 0.5 = -7.544313532 Reorder the terms: 0.5 + x = -7.544313532 Solving 0.5 + x = -7.544313532 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 = -7.544313532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.544313532 + -0.5 x = -7.544313532 + -0.5 Combine like terms: -7.544313532 + -0.5 = -8.044313532 x = -8.044313532 Simplifying x = -8.044313532Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.044313532, -8.044313532}
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