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Simplifying 2x2 + x + -40 = 180 Reorder the terms: -40 + x + 2x2 = 180 Solving -40 + x + 2x2 = 180 Solving for variable 'x'. Reorder the terms: -40 + -180 + x + 2x2 = 180 + -180 Combine like terms: -40 + -180 = -220 -220 + x + 2x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -220 + x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -110 + 0.5x + x2 = 0 Move the constant term to the right: Add '110' to each side of the equation. -110 + 0.5x + 110 + x2 = 0 + 110 Reorder the terms: -110 + 110 + 0.5x + x2 = 0 + 110 Combine like terms: -110 + 110 = 0 0 + 0.5x + x2 = 0 + 110 0.5x + x2 = 0 + 110 Combine like terms: 0 + 110 = 110 0.5x + x2 = 110 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. 0.5x + 0.25 + x2 = 110 + 0.25 Reorder the terms: 0.25 + 0.5x + x2 = 110 + 0.25 Combine like terms: 110 + 0.25 = 110.25 0.25 + 0.5x + x2 = 110.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 110.25 Calculate the square root of the right side: 10.5 Break this problem into two subproblems by setting (x + 0.5) equal to 10.5 and -10.5.Subproblem 1
x + 0.5 = 10.5 Simplifying x + 0.5 = 10.5 Reorder the terms: 0.5 + x = 10.5 Solving 0.5 + x = 10.5 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 = 10.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.5 + -0.5 x = 10.5 + -0.5 Combine like terms: 10.5 + -0.5 = 10 x = 10 Simplifying x = 10Subproblem 2
x + 0.5 = -10.5 Simplifying x + 0.5 = -10.5 Reorder the terms: 0.5 + x = -10.5 Solving 0.5 + x = -10.5 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 = -10.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.5 + -0.5 x = -10.5 + -0.5 Combine like terms: -10.5 + -0.5 = -11 x = -11 Simplifying x = -11Solution
The solution to the problem is based on the solutions from the subproblems. x = {10, -11}
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