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Simplifying 100a2 + -10a + -3000 = 0 Reorder the terms: -3000 + -10a + 100a2 = 0 Solving -3000 + -10a + 100a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '10'. 10(-300 + -1a + 10a2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-300 + -1a + 10a2)' equal to zero and attempt to solve: Simplifying -300 + -1a + 10a2 = 0 Solving -300 + -1a + 10a2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -30 + -0.1a + a2 = 0.0 Move the constant term to the right: Add '30' to each side of the equation. -30 + -0.1a + 30 + a2 = 0.0 + 30 Reorder the terms: -30 + 30 + -0.1a + a2 = 0.0 + 30 Combine like terms: -30 + 30 = 0 0 + -0.1a + a2 = 0.0 + 30 -0.1a + a2 = 0.0 + 30 Combine like terms: 0.0 + 30 = 30 -0.1a + a2 = 30 The a term is -0.1a. Take half its coefficient (-0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. -0.1a + 0.0025 + a2 = 30 + 0.0025 Reorder the terms: 0.0025 + -0.1a + a2 = 30 + 0.0025 Combine like terms: 30 + 0.0025 = 30.0025 0.0025 + -0.1a + a2 = 30.0025 Factor a perfect square on the left side: (a + -0.05)(a + -0.05) = 30.0025 Calculate the square root of the right side: 5.477453788 Break this problem into two subproblems by setting (a + -0.05) equal to 5.477453788 and -5.477453788.Subproblem 1
a + -0.05 = 5.477453788 Simplifying a + -0.05 = 5.477453788 Reorder the terms: -0.05 + a = 5.477453788 Solving -0.05 + a = 5.477453788 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + a = 5.477453788 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + a = 5.477453788 + 0.05 a = 5.477453788 + 0.05 Combine like terms: 5.477453788 + 0.05 = 5.527453788 a = 5.527453788 Simplifying a = 5.527453788Subproblem 2
a + -0.05 = -5.477453788 Simplifying a + -0.05 = -5.477453788 Reorder the terms: -0.05 + a = -5.477453788 Solving -0.05 + a = -5.477453788 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + a = -5.477453788 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + a = -5.477453788 + 0.05 a = -5.477453788 + 0.05 Combine like terms: -5.477453788 + 0.05 = -5.427453788 a = -5.427453788 Simplifying a = -5.427453788Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.527453788, -5.427453788}Solution
a = {5.527453788, -5.427453788}
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