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Simplifying 2x2 + 3x = 105 Reorder the terms: 3x + 2x2 = 105 Solving 3x + 2x2 = 105 Solving for variable 'x'. Reorder the terms: -105 + 3x + 2x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -105 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -52.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '52.5' to each side of the equation. -52.5 + 1.5x + 52.5 + x2 = 0 + 52.5 Reorder the terms: -52.5 + 52.5 + 1.5x + x2 = 0 + 52.5 Combine like terms: -52.5 + 52.5 = 0.0 0.0 + 1.5x + x2 = 0 + 52.5 1.5x + x2 = 0 + 52.5 Combine like terms: 0 + 52.5 = 52.5 1.5x + x2 = 52.5 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 = 52.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 52.5 + 0.5625 Combine like terms: 52.5 + 0.5625 = 53.0625 0.5625 + 1.5x + x2 = 53.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 53.0625 Calculate the square root of the right side: 7.284401142 Break this problem into two subproblems by setting (x + 0.75) equal to 7.284401142 and -7.284401142.Subproblem 1
x + 0.75 = 7.284401142 Simplifying x + 0.75 = 7.284401142 Reorder the terms: 0.75 + x = 7.284401142 Solving 0.75 + x = 7.284401142 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 = 7.284401142 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 7.284401142 + -0.75 x = 7.284401142 + -0.75 Combine like terms: 7.284401142 + -0.75 = 6.534401142 x = 6.534401142 Simplifying x = 6.534401142Subproblem 2
x + 0.75 = -7.284401142 Simplifying x + 0.75 = -7.284401142 Reorder the terms: 0.75 + x = -7.284401142 Solving 0.75 + x = -7.284401142 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 = -7.284401142 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -7.284401142 + -0.75 x = -7.284401142 + -0.75 Combine like terms: -7.284401142 + -0.75 = -8.034401142 x = -8.034401142 Simplifying x = -8.034401142Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.534401142, -8.034401142}
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