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Simplifying 6x2 + 3x + 4 = 2010 Reorder the terms: 4 + 3x + 6x2 = 2010 Solving 4 + 3x + 6x2 = 2010 Solving for variable 'x'. Reorder the terms: 4 + -2010 + 3x + 6x2 = 2010 + -2010 Combine like terms: 4 + -2010 = -2006 -2006 + 3x + 6x2 = 2010 + -2010 Combine like terms: 2010 + -2010 = 0 -2006 + 3x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -334.3333333 + 0.5x + x2 = 0 Move the constant term to the right: Add '334.3333333' to each side of the equation. -334.3333333 + 0.5x + 334.3333333 + x2 = 0 + 334.3333333 Reorder the terms: -334.3333333 + 334.3333333 + 0.5x + x2 = 0 + 334.3333333 Combine like terms: -334.3333333 + 334.3333333 = 0.0000000 0.0000000 + 0.5x + x2 = 0 + 334.3333333 0.5x + x2 = 0 + 334.3333333 Combine like terms: 0 + 334.3333333 = 334.3333333 0.5x + x2 = 334.3333333 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 = 334.3333333 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 334.3333333 + 0.0625 Combine like terms: 334.3333333 + 0.0625 = 334.3958333 0.0625 + 0.5x + x2 = 334.3958333 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 334.3958333 Calculate the square root of the right side: 18.286493193 Break this problem into two subproblems by setting (x + 0.25) equal to 18.286493193 and -18.286493193.Subproblem 1
x + 0.25 = 18.286493193 Simplifying x + 0.25 = 18.286493193 Reorder the terms: 0.25 + x = 18.286493193 Solving 0.25 + x = 18.286493193 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 = 18.286493193 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 18.286493193 + -0.25 x = 18.286493193 + -0.25 Combine like terms: 18.286493193 + -0.25 = 18.036493193 x = 18.036493193 Simplifying x = 18.036493193Subproblem 2
x + 0.25 = -18.286493193 Simplifying x + 0.25 = -18.286493193 Reorder the terms: 0.25 + x = -18.286493193 Solving 0.25 + x = -18.286493193 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 = -18.286493193 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -18.286493193 + -0.25 x = -18.286493193 + -0.25 Combine like terms: -18.286493193 + -0.25 = -18.536493193 x = -18.536493193 Simplifying x = -18.536493193Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.036493193, -18.536493193}
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