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Simplifying 5x(-4x + 4) = 1 Reorder the terms: 5x(4 + -4x) = 1 (4 * 5x + -4x * 5x) = 1 (20x + -20x2) = 1 Solving 20x + -20x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 20x + -20x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 20x + -20x2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. 0.05 + -1x + x2 = 0.0 Move the constant term to the right: Add '-0.05' to each side of the equation. 0.05 + -1x + -0.05 + x2 = 0.0 + -0.05 Reorder the terms: 0.05 + -0.05 + -1x + x2 = 0.0 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + -1x + x2 = 0.0 + -0.05 -1x + x2 = 0.0 + -0.05 Combine like terms: 0.0 + -0.05 = -0.05 -1x + x2 = -0.05 The x term is -1x. 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. -1x + 0.25 + x2 = -0.05 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.05 + 0.25 Combine like terms: -0.05 + 0.25 = 0.2 0.25 + -1x + x2 = 0.2 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.2 Calculate the square root of the right side: 0.447213596 Break this problem into two subproblems by setting (x + -0.5) equal to 0.447213596 and -0.447213596.Subproblem 1
x + -0.5 = 0.447213596 Simplifying x + -0.5 = 0.447213596 Reorder the terms: -0.5 + x = 0.447213596 Solving -0.5 + x = 0.447213596 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 = 0.447213596 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.447213596 + 0.5 x = 0.447213596 + 0.5 Combine like terms: 0.447213596 + 0.5 = 0.947213596 x = 0.947213596 Simplifying x = 0.947213596Subproblem 2
x + -0.5 = -0.447213596 Simplifying x + -0.5 = -0.447213596 Reorder the terms: -0.5 + x = -0.447213596 Solving -0.5 + x = -0.447213596 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 = -0.447213596 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.447213596 + 0.5 x = -0.447213596 + 0.5 Combine like terms: -0.447213596 + 0.5 = 0.052786404 x = 0.052786404 Simplifying x = 0.052786404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.947213596, 0.052786404}
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