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Simplifying 0 = 5x2 + -10x + -5 Reorder the terms: 0 = -5 + -10x + 5x2 Solving 0 = -5 + -10x + 5x2 Solving for variable 'x'. Combine like terms: 0 + 5 = 5 5 + 10x + -5x2 = -5 + -10x + 5x2 + 5 + 10x + -5x2 Reorder the terms: 5 + 10x + -5x2 = -5 + 5 + -10x + 10x + 5x2 + -5x2 Combine like terms: -5 + 5 = 0 5 + 10x + -5x2 = 0 + -10x + 10x + 5x2 + -5x2 5 + 10x + -5x2 = -10x + 10x + 5x2 + -5x2 Combine like terms: -10x + 10x = 0 5 + 10x + -5x2 = 0 + 5x2 + -5x2 5 + 10x + -5x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 5 + 10x + -5x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(1 + 2x + -1x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(1 + 2x + -1x2)' equal to zero and attempt to solve: Simplifying 1 + 2x + -1x2 = 0 Solving 1 + 2x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + -2x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -2x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + -2x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2x + x2 = 0 + 1 -2x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2x + x2 = 1 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 = 1 + 1 Reorder the terms: 1 + -2x + x2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + -2x + x2 = 2 Factor a perfect square on the left side: (x + -1)(x + -1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (x + -1) equal to 1.414213562 and -1.414213562.Subproblem 1
x + -1 = 1.414213562 Simplifying x + -1 = 1.414213562 Reorder the terms: -1 + x = 1.414213562 Solving -1 + x = 1.414213562 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 = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.414213562 + 1 x = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 x = 2.414213562 Simplifying x = 2.414213562Subproblem 2
x + -1 = -1.414213562 Simplifying x + -1 = -1.414213562 Reorder the terms: -1 + x = -1.414213562 Solving -1 + x = -1.414213562 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 = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.414213562 + 1 x = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 x = -0.414213562 Simplifying x = -0.414213562Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.414213562, -0.414213562}Solution
x = {2.414213562, -0.414213562}
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