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Simplifying a(a + -5) = 32 Reorder the terms: a(-5 + a) = 32 (-5 * a + a * a) = 32 (-5a + a2) = 32 Solving -5a + a2 = 32 Solving for variable 'a'. Reorder the terms: -32 + -5a + a2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + -5a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + -5a + 32 + a2 = 0 + 32 Reorder the terms: -32 + 32 + -5a + a2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -5a + a2 = 0 + 32 -5a + a2 = 0 + 32 Combine like terms: 0 + 32 = 32 -5a + a2 = 32 The a term is -5a. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5a + 6.25 + a2 = 32 + 6.25 Reorder the terms: 6.25 + -5a + a2 = 32 + 6.25 Combine like terms: 32 + 6.25 = 38.25 6.25 + -5a + a2 = 38.25 Factor a perfect square on the left side: (a + -2.5)(a + -2.5) = 38.25 Calculate the square root of the right side: 6.184658438 Break this problem into two subproblems by setting (a + -2.5) equal to 6.184658438 and -6.184658438.Subproblem 1
a + -2.5 = 6.184658438 Simplifying a + -2.5 = 6.184658438 Reorder the terms: -2.5 + a = 6.184658438 Solving -2.5 + a = 6.184658438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = 6.184658438 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = 6.184658438 + 2.5 a = 6.184658438 + 2.5 Combine like terms: 6.184658438 + 2.5 = 8.684658438 a = 8.684658438 Simplifying a = 8.684658438Subproblem 2
a + -2.5 = -6.184658438 Simplifying a + -2.5 = -6.184658438 Reorder the terms: -2.5 + a = -6.184658438 Solving -2.5 + a = -6.184658438 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + a = -6.184658438 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + a = -6.184658438 + 2.5 a = -6.184658438 + 2.5 Combine like terms: -6.184658438 + 2.5 = -3.684658438 a = -3.684658438 Simplifying a = -3.684658438Solution
The solution to the problem is based on the solutions from the subproblems. a = {8.684658438, -3.684658438}
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