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Simplifying 4x2 + 1 = 8x Reorder the terms: 1 + 4x2 = 8x Solving 1 + 4x2 = 8x Solving for variable 'x'. Reorder the terms: 1 + -8x + 4x2 = 8x + -8x Combine like terms: 8x + -8x = 0 1 + -8x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -2x + x2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -2x + -0.25 + x2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -2x + x2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -2x + x2 = 0 + -0.25 -2x + x2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -2x + x2 = -0.25 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.25 + 1 Reorder the terms: 1 + -2x + x2 = -0.25 + 1 Combine like terms: -0.25 + 1 = 0.75 1 + -2x + x2 = 0.75 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + -1) equal to 0.866025404 and -0.866025404.Subproblem 1
x + -1 = 0.866025404 Simplifying x + -1 = 0.866025404 Reorder the terms: -1 + x = 0.866025404 Solving -1 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.866025404 + 1 x = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 x = 1.866025404 Simplifying x = 1.866025404Subproblem 2
x + -1 = -0.866025404 Simplifying x + -1 = -0.866025404 Reorder the terms: -1 + x = -0.866025404 Solving -1 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.866025404 + 1 x = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 x = 0.133974596 Simplifying x = 0.133974596Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.866025404, 0.133974596}
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