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Simplifying 75 = (10000 + 100x) * x Reorder the terms for easier multiplication: 75 = x(10000 + 100x) 75 = (10000 * x + 100x * x) 75 = (10000x + 100x2) Solving 75 = 10000x + 100x2 Solving for variable 'x'. Reorder the terms: 75 + -10000x + -100x2 = 10000x + -10000x + 100x2 + -100x2 Combine like terms: 10000x + -10000x = 0 75 + -10000x + -100x2 = 0 + 100x2 + -100x2 75 + -10000x + -100x2 = 100x2 + -100x2 Combine like terms: 100x2 + -100x2 = 0 75 + -10000x + -100x2 = 0 Factor out the Greatest Common Factor (GCF), '25'. 25(3 + -400x + -4x2) = 0 Ignore the factor 25.Subproblem 1
Set the factor '(3 + -400x + -4x2)' equal to zero and attempt to solve: Simplifying 3 + -400x + -4x2 = 0 Solving 3 + -400x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + 100x + x2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 100x + 0.75 + x2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 100x + x2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 100x + x2 = 0 + 0.75 100x + x2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 100x + x2 = 0.75 The x term is 100x. Take half its coefficient (50). Square it (2500) and add it to both sides. Add '2500' to each side of the equation. 100x + 2500 + x2 = 0.75 + 2500 Reorder the terms: 2500 + 100x + x2 = 0.75 + 2500 Combine like terms: 0.75 + 2500 = 2500.75 2500 + 100x + x2 = 2500.75 Factor a perfect square on the left side: (x + 50)(x + 50) = 2500.75 Calculate the square root of the right side: 50.007499438 Break this problem into two subproblems by setting (x + 50) equal to 50.007499438 and -50.007499438.Subproblem 1
x + 50 = 50.007499438 Simplifying x + 50 = 50.007499438 Reorder the terms: 50 + x = 50.007499438 Solving 50 + x = 50.007499438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-50' to each side of the equation. 50 + -50 + x = 50.007499438 + -50 Combine like terms: 50 + -50 = 0 0 + x = 50.007499438 + -50 x = 50.007499438 + -50 Combine like terms: 50.007499438 + -50 = 0.007499438 x = 0.007499438 Simplifying x = 0.007499438Subproblem 2
x + 50 = -50.007499438 Simplifying x + 50 = -50.007499438 Reorder the terms: 50 + x = -50.007499438 Solving 50 + x = -50.007499438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-50' to each side of the equation. 50 + -50 + x = -50.007499438 + -50 Combine like terms: 50 + -50 = 0 0 + x = -50.007499438 + -50 x = -50.007499438 + -50 Combine like terms: -50.007499438 + -50 = -100.007499438 x = -100.007499438 Simplifying x = -100.007499438Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.007499438, -100.007499438}Solution
x = {0.007499438, -100.007499438}
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