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Simplifying 0 = 5x2 + 5x + 1 Reorder the terms: 0 = 1 + 5x + 5x2 Solving 0 = 1 + 5x + 5x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + -5x + -5x2 = 1 + 5x + 5x2 + -1 + -5x + -5x2 Reorder the terms: -1 + -5x + -5x2 = 1 + -1 + 5x + -5x + 5x2 + -5x2 Combine like terms: 1 + -1 = 0 -1 + -5x + -5x2 = 0 + 5x + -5x + 5x2 + -5x2 -1 + -5x + -5x2 = 5x + -5x + 5x2 + -5x2 Combine like terms: 5x + -5x = 0 -1 + -5x + -5x2 = 0 + 5x2 + -5x2 -1 + -5x + -5x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 -1 + -5x + -5x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 5x + 5x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 5x + 5x2)' equal to zero and attempt to solve: Simplifying 1 + 5x + 5x2 = 0 Solving 1 + 5x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + x + x2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + x + -0.2 + x2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + x + x2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + x + x2 = 0 + -0.2 x + x2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 x + x2 = -0.2 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 = -0.2 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = -0.2 + 0.25 Combine like terms: -0.2 + 0.25 = 0.05 1.25 + x2 = 0.05 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (x + 0.5) equal to 0.223606798 and -0.223606798.Subproblem 1
x + 0.5 = 0.223606798 Simplifying x + 0.5 = 0.223606798 Reorder the terms: 0.5 + x = 0.223606798 Solving 0.5 + x = 0.223606798 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 = 0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.223606798 + -0.5 x = 0.223606798 + -0.5 Combine like terms: 0.223606798 + -0.5 = -0.276393202 x = -0.276393202 Simplifying x = -0.276393202Subproblem 2
x + 0.5 = -0.223606798 Simplifying x + 0.5 = -0.223606798 Reorder the terms: 0.5 + x = -0.223606798 Solving 0.5 + x = -0.223606798 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 = -0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.223606798 + -0.5 x = -0.223606798 + -0.5 Combine like terms: -0.223606798 + -0.5 = -0.723606798 x = -0.723606798 Simplifying x = -0.723606798Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.276393202, -0.723606798}Solution
x = {-0.276393202, -0.723606798}
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