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Simplifying 4x2 + 4x + 1 = 500000 Reorder the terms: 1 + 4x + 4x2 = 500000 Solving 1 + 4x + 4x2 = 500000 Solving for variable 'x'. Reorder the terms: 1 + -500000 + 4x + 4x2 = 500000 + -500000 Combine like terms: 1 + -500000 = -499999 -499999 + 4x + 4x2 = 500000 + -500000 Combine like terms: 500000 + -500000 = 0 -499999 + 4x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -124999.75 + x + x2 = 0 Move the constant term to the right: Add '124999.75' to each side of the equation. -124999.75 + x + 124999.75 + x2 = 0 + 124999.75 Reorder the terms: -124999.75 + 124999.75 + x + x2 = 0 + 124999.75 Combine like terms: -124999.75 + 124999.75 = 0.00 0.00 + x + x2 = 0 + 124999.75 x + x2 = 0 + 124999.75 Combine like terms: 0 + 124999.75 = 124999.75 x + x2 = 124999.75 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. + 0.25 + x2 = 124999.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 124999.75 + 0.25 Combine like terms: 124999.75 + 0.25 = 125000 1.25 + x2 = 125000 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 125000 Calculate the square root of the right side: 353.553390593 Break this problem into two subproblems by setting (x + 0.5) equal to 353.553390593 and -353.553390593.Subproblem 1
x + 0.5 = 353.553390593 Simplifying x + 0.5 = 353.553390593 Reorder the terms: 0.5 + x = 353.553390593 Solving 0.5 + x = 353.553390593 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 = 353.553390593 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 353.553390593 + -0.5 x = 353.553390593 + -0.5 Combine like terms: 353.553390593 + -0.5 = 353.053390593 x = 353.053390593 Simplifying x = 353.053390593Subproblem 2
x + 0.5 = -353.553390593 Simplifying x + 0.5 = -353.553390593 Reorder the terms: 0.5 + x = -353.553390593 Solving 0.5 + x = -353.553390593 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 = -353.553390593 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -353.553390593 + -0.5 x = -353.553390593 + -0.5 Combine like terms: -353.553390593 + -0.5 = -354.053390593 x = -354.053390593 Simplifying x = -354.053390593Solution
The solution to the problem is based on the solutions from the subproblems. x = {353.053390593, -354.053390593}
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