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Simplifying 2x2 + 3x + 1 = 1990 Reorder the terms: 1 + 3x + 2x2 = 1990 Solving 1 + 3x + 2x2 = 1990 Solving for variable 'x'. Reorder the terms: 1 + -1990 + 3x + 2x2 = 1990 + -1990 Combine like terms: 1 + -1990 = -1989 -1989 + 3x + 2x2 = 1990 + -1990 Combine like terms: 1990 + -1990 = 0 -1989 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -994.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '994.5' to each side of the equation. -994.5 + 1.5x + 994.5 + x2 = 0 + 994.5 Reorder the terms: -994.5 + 994.5 + 1.5x + x2 = 0 + 994.5 Combine like terms: -994.5 + 994.5 = 0.0 0.0 + 1.5x + x2 = 0 + 994.5 1.5x + x2 = 0 + 994.5 Combine like terms: 0 + 994.5 = 994.5 1.5x + x2 = 994.5 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 994.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 994.5 + 0.5625 Combine like terms: 994.5 + 0.5625 = 995.0625 0.5625 + 1.5x + x2 = 995.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 995.0625 Calculate the square root of the right side: 31.544611267 Break this problem into two subproblems by setting (x + 0.75) equal to 31.544611267 and -31.544611267.Subproblem 1
x + 0.75 = 31.544611267 Simplifying x + 0.75 = 31.544611267 Reorder the terms: 0.75 + x = 31.544611267 Solving 0.75 + x = 31.544611267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 31.544611267 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 31.544611267 + -0.75 x = 31.544611267 + -0.75 Combine like terms: 31.544611267 + -0.75 = 30.794611267 x = 30.794611267 Simplifying x = 30.794611267Subproblem 2
x + 0.75 = -31.544611267 Simplifying x + 0.75 = -31.544611267 Reorder the terms: 0.75 + x = -31.544611267 Solving 0.75 + x = -31.544611267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -31.544611267 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -31.544611267 + -0.75 x = -31.544611267 + -0.75 Combine like terms: -31.544611267 + -0.75 = -32.294611267 x = -32.294611267 Simplifying x = -32.294611267Solution
The solution to the problem is based on the solutions from the subproblems. x = {30.794611267, -32.294611267}
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