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Simplifying 4x2 + 1 = 2 + -5x Reorder the terms: 1 + 4x2 = 2 + -5x Solving 1 + 4x2 = 2 + -5x Solving for variable 'x'. Reorder the terms: 1 + -2 + 5x + 4x2 = 2 + -5x + -2 + 5x Combine like terms: 1 + -2 = -1 -1 + 5x + 4x2 = 2 + -5x + -2 + 5x Reorder the terms: -1 + 5x + 4x2 = 2 + -2 + -5x + 5x Combine like terms: 2 + -2 = 0 -1 + 5x + 4x2 = 0 + -5x + 5x -1 + 5x + 4x2 = -5x + 5x Combine like terms: -5x + 5x = 0 -1 + 5x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + 1.25x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 1.25x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 1.25x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 1.25x + x2 = 0 + 0.25 1.25x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 1.25x + x2 = 0.25 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 = 0.25 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + 1.25x + x2 = 0.640625 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (x + 0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
x + 0.625 = 0.80039053 Simplifying x + 0.625 = 0.80039053 Reorder the terms: 0.625 + x = 0.80039053 Solving 0.625 + x = 0.80039053 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 = 0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 0.80039053 + -0.625 x = 0.80039053 + -0.625 Combine like terms: 0.80039053 + -0.625 = 0.17539053 x = 0.17539053 Simplifying x = 0.17539053Subproblem 2
x + 0.625 = -0.80039053 Simplifying x + 0.625 = -0.80039053 Reorder the terms: 0.625 + x = -0.80039053 Solving 0.625 + x = -0.80039053 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 = -0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -0.80039053 + -0.625 x = -0.80039053 + -0.625 Combine like terms: -0.80039053 + -0.625 = -1.42539053 x = -1.42539053 Simplifying x = -1.42539053Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.17539053, -1.42539053}
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