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Simplifying 0.2x + 1 = -0.4x2 + 12 Reorder the terms: 1 + 0.2x = -0.4x2 + 12 Reorder the terms: 1 + 0.2x = 12 + -0.4x2 Solving 1 + 0.2x = 12 + -0.4x2 Solving for variable 'x'. Reorder the terms: 1 + -12 + 0.2x + 0.4x2 = 12 + -0.4x2 + -12 + 0.4x2 Combine like terms: 1 + -12 = -11 -11 + 0.2x + 0.4x2 = 12 + -0.4x2 + -12 + 0.4x2 Reorder the terms: -11 + 0.2x + 0.4x2 = 12 + -12 + -0.4x2 + 0.4x2 Combine like terms: 12 + -12 = 0 -11 + 0.2x + 0.4x2 = 0 + -0.4x2 + 0.4x2 -11 + 0.2x + 0.4x2 = -0.4x2 + 0.4x2 Combine like terms: -0.4x2 + 0.4x2 = 0.0 -11 + 0.2x + 0.4x2 = 0.0 Begin completing the square. Divide all terms by 0.4 the coefficient of the squared term: Divide each side by '0.4'. -27.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '27.5' to each side of the equation. -27.5 + 0.5x + 27.5 + x2 = 0 + 27.5 Reorder the terms: -27.5 + 27.5 + 0.5x + x2 = 0 + 27.5 Combine like terms: -27.5 + 27.5 = 0.0 0.0 + 0.5x + x2 = 0 + 27.5 0.5x + x2 = 0 + 27.5 Combine like terms: 0 + 27.5 = 27.5 0.5x + x2 = 27.5 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 = 27.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 27.5 + 0.0625 Combine like terms: 27.5 + 0.0625 = 27.5625 0.0625 + 0.5x + x2 = 27.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 27.5625 Calculate the square root of the right side: 5.25 Break this problem into two subproblems by setting (x + 0.25) equal to 5.25 and -5.25.Subproblem 1
x + 0.25 = 5.25 Simplifying x + 0.25 = 5.25 Reorder the terms: 0.25 + x = 5.25 Solving 0.25 + x = 5.25 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 = 5.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.25 + -0.25 x = 5.25 + -0.25 Combine like terms: 5.25 + -0.25 = 5 x = 5 Simplifying x = 5Subproblem 2
x + 0.25 = -5.25 Simplifying x + 0.25 = -5.25 Reorder the terms: 0.25 + x = -5.25 Solving 0.25 + x = -5.25 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 = -5.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.25 + -0.25 x = -5.25 + -0.25 Combine like terms: -5.25 + -0.25 = -5.5 x = -5.5 Simplifying x = -5.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {5, -5.5}
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