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Simplifying 1x(40 + 440x) = 1000 (40 * 1x + 440x * 1x) = 1000 (40x + 440x2) = 1000 Solving 40x + 440x2 = 1000 Solving for variable 'x'. Reorder the terms: -1000 + 40x + 440x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 40x + 440x2 = 0 Factor out the Greatest Common Factor (GCF), '40'. 40(-25 + x + 11x2) = 0 Ignore the factor 40.Subproblem 1
Set the factor '(-25 + x + 11x2)' equal to zero and attempt to solve: Simplifying -25 + x + 11x2 = 0 Solving -25 + x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.272727273 + 0.09090909091x + x2 = 0 Move the constant term to the right: Add '2.272727273' to each side of the equation. -2.272727273 + 0.09090909091x + 2.272727273 + x2 = 0 + 2.272727273 Reorder the terms: -2.272727273 + 2.272727273 + 0.09090909091x + x2 = 0 + 2.272727273 Combine like terms: -2.272727273 + 2.272727273 = 0.000000000 0.000000000 + 0.09090909091x + x2 = 0 + 2.272727273 0.09090909091x + x2 = 0 + 2.272727273 Combine like terms: 0 + 2.272727273 = 2.272727273 0.09090909091x + x2 = 2.272727273 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. 0.09090909091x + 0.25 + x2 = 2.272727273 + 0.25 Reorder the terms: 0.25 + 0.09090909091x + x2 = 2.272727273 + 0.25 Combine like terms: 2.272727273 + 0.25 = 2.522727273 0.25 + 0.09090909091x + x2 = 2.522727273 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 2.522727273 Calculate the square root of the right side: 1.588309565 Break this problem into two subproblems by setting (x + 0.5) equal to 1.588309565 and -1.588309565.Subproblem 1
x + 0.5 = 1.588309565 Simplifying x + 0.5 = 1.588309565 Reorder the terms: 0.5 + x = 1.588309565 Solving 0.5 + x = 1.588309565 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 = 1.588309565 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.588309565 + -0.5 x = 1.588309565 + -0.5 Combine like terms: 1.588309565 + -0.5 = 1.088309565 x = 1.088309565 Simplifying x = 1.088309565Subproblem 2
x + 0.5 = -1.588309565 Simplifying x + 0.5 = -1.588309565 Reorder the terms: 0.5 + x = -1.588309565 Solving 0.5 + x = -1.588309565 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 = -1.588309565 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.588309565 + -0.5 x = -1.588309565 + -0.5 Combine like terms: -1.588309565 + -0.5 = -2.088309565 x = -2.088309565 Simplifying x = -2.088309565Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.088309565, -2.088309565}Solution
x = {1.088309565, -2.088309565}
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