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Simplifying 82x2 + 10x + -24 = 0 Reorder the terms: -24 + 10x + 82x2 = 0 Solving -24 + 10x + 82x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + 5x + 41x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + 5x + 41x2)' equal to zero and attempt to solve: Simplifying -12 + 5x + 41x2 = 0 Solving -12 + 5x + 41x2 = 0 Begin completing the square. Divide all terms by 41 the coefficient of the squared term: Divide each side by '41'. -0.2926829268 + 0.1219512195x + x2 = 0 Move the constant term to the right: Add '0.2926829268' to each side of the equation. -0.2926829268 + 0.1219512195x + 0.2926829268 + x2 = 0 + 0.2926829268 Reorder the terms: -0.2926829268 + 0.2926829268 + 0.1219512195x + x2 = 0 + 0.2926829268 Combine like terms: -0.2926829268 + 0.2926829268 = 0.0000000000 0.0000000000 + 0.1219512195x + x2 = 0 + 0.2926829268 0.1219512195x + x2 = 0 + 0.2926829268 Combine like terms: 0 + 0.2926829268 = 0.2926829268 0.1219512195x + x2 = 0.2926829268 The x term is 0.1219512195x. Take half its coefficient (0.06097560975). Square it (0.003718024984) and add it to both sides. Add '0.003718024984' to each side of the equation. 0.1219512195x + 0.003718024984 + x2 = 0.2926829268 + 0.003718024984 Reorder the terms: 0.003718024984 + 0.1219512195x + x2 = 0.2926829268 + 0.003718024984 Combine like terms: 0.2926829268 + 0.003718024984 = 0.296400951784 0.003718024984 + 0.1219512195x + x2 = 0.296400951784 Factor a perfect square on the left side: (x + 0.06097560975)(x + 0.06097560975) = 0.296400951784 Calculate the square root of the right side: 0.544427178 Break this problem into two subproblems by setting (x + 0.06097560975) equal to 0.544427178 and -0.544427178.Subproblem 1
x + 0.06097560975 = 0.544427178 Simplifying x + 0.06097560975 = 0.544427178 Reorder the terms: 0.06097560975 + x = 0.544427178 Solving 0.06097560975 + x = 0.544427178 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.06097560975' to each side of the equation. 0.06097560975 + -0.06097560975 + x = 0.544427178 + -0.06097560975 Combine like terms: 0.06097560975 + -0.06097560975 = 0.00000000000 0.00000000000 + x = 0.544427178 + -0.06097560975 x = 0.544427178 + -0.06097560975 Combine like terms: 0.544427178 + -0.06097560975 = 0.48345156825 x = 0.48345156825 Simplifying x = 0.48345156825Subproblem 2
x + 0.06097560975 = -0.544427178 Simplifying x + 0.06097560975 = -0.544427178 Reorder the terms: 0.06097560975 + x = -0.544427178 Solving 0.06097560975 + x = -0.544427178 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.06097560975' to each side of the equation. 0.06097560975 + -0.06097560975 + x = -0.544427178 + -0.06097560975 Combine like terms: 0.06097560975 + -0.06097560975 = 0.00000000000 0.00000000000 + x = -0.544427178 + -0.06097560975 x = -0.544427178 + -0.06097560975 Combine like terms: -0.544427178 + -0.06097560975 = -0.60540278775 x = -0.60540278775 Simplifying x = -0.60540278775Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.48345156825, -0.60540278775}Solution
x = {0.48345156825, -0.60540278775}
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