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Simplifying 2a(a + 10) * 44 = 180 Reorder the terms: 2a(10 + a) * 44 = 180 Reorder the terms for easier multiplication: 2 * 44a(10 + a) = 180 Multiply 2 * 44 88a(10 + a) = 180 (10 * 88a + a * 88a) = 180 (880a + 88a2) = 180 Solving 880a + 88a2 = 180 Solving for variable 'a'. Reorder the terms: -180 + 880a + 88a2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 880a + 88a2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-45 + 220a + 22a2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-45 + 220a + 22a2)' equal to zero and attempt to solve: Simplifying -45 + 220a + 22a2 = 0 Solving -45 + 220a + 22a2 = 0 Begin completing the square. Divide all terms by 22 the coefficient of the squared term: Divide each side by '22'. -2.045454545 + 10a + a2 = 0 Move the constant term to the right: Add '2.045454545' to each side of the equation. -2.045454545 + 10a + 2.045454545 + a2 = 0 + 2.045454545 Reorder the terms: -2.045454545 + 2.045454545 + 10a + a2 = 0 + 2.045454545 Combine like terms: -2.045454545 + 2.045454545 = 0.000000000 0.000000000 + 10a + a2 = 0 + 2.045454545 10a + a2 = 0 + 2.045454545 Combine like terms: 0 + 2.045454545 = 2.045454545 10a + a2 = 2.045454545 The a term is 10a. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10a + 25 + a2 = 2.045454545 + 25 Reorder the terms: 25 + 10a + a2 = 2.045454545 + 25 Combine like terms: 2.045454545 + 25 = 27.045454545 25 + 10a + a2 = 27.045454545 Factor a perfect square on the left side: (a + 5)(a + 5) = 27.045454545 Calculate the square root of the right side: 5.200524449 Break this problem into two subproblems by setting (a + 5) equal to 5.200524449 and -5.200524449.Subproblem 1
a + 5 = 5.200524449 Simplifying a + 5 = 5.200524449 Reorder the terms: 5 + a = 5.200524449 Solving 5 + a = 5.200524449 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + a = 5.200524449 + -5 Combine like terms: 5 + -5 = 0 0 + a = 5.200524449 + -5 a = 5.200524449 + -5 Combine like terms: 5.200524449 + -5 = 0.200524449 a = 0.200524449 Simplifying a = 0.200524449Subproblem 2
a + 5 = -5.200524449 Simplifying a + 5 = -5.200524449 Reorder the terms: 5 + a = -5.200524449 Solving 5 + a = -5.200524449 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + a = -5.200524449 + -5 Combine like terms: 5 + -5 = 0 0 + a = -5.200524449 + -5 a = -5.200524449 + -5 Combine like terms: -5.200524449 + -5 = -10.200524449 a = -10.200524449 Simplifying a = -10.200524449Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.200524449, -10.200524449}Solution
a = {0.200524449, -10.200524449}
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