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Simplifying 4x2 + 5x + -0.5 = 0 Reorder the terms: -0.5 + 5x + 4x2 = 0 Solving -0.5 + 5x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.125 + 1.25x + x2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 1.25x + 0.125 + x2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 1.25x + x2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 1.25x + x2 = 0 + 0.125 1.25x + x2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 1.25x + x2 = 0.125 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = 0.125 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 0.125 + 0.390625 Combine like terms: 0.125 + 0.390625 = 0.515625 0.390625 + 1.25x + x2 = 0.515625 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (x + 0.625) equal to 0.718070331 and -0.718070331.Subproblem 1
x + 0.625 = 0.718070331 Simplifying x + 0.625 = 0.718070331 Reorder the terms: 0.625 + x = 0.718070331 Solving 0.625 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 0.718070331 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 0.718070331 + -0.625 x = 0.718070331 + -0.625 Combine like terms: 0.718070331 + -0.625 = 0.093070331 x = 0.093070331 Simplifying x = 0.093070331Subproblem 2
x + 0.625 = -0.718070331 Simplifying x + 0.625 = -0.718070331 Reorder the terms: 0.625 + x = -0.718070331 Solving 0.625 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -0.718070331 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -0.718070331 + -0.625 x = -0.718070331 + -0.625 Combine like terms: -0.718070331 + -0.625 = -1.343070331 x = -1.343070331 Simplifying x = -1.343070331Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.093070331, -1.343070331}
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