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Simplifying 19a(a + -6) = 69 Reorder the terms: 19a(-6 + a) = 69 (-6 * 19a + a * 19a) = 69 (-114a + 19a2) = 69 Solving -114a + 19a2 = 69 Solving for variable 'a'. Reorder the terms: -69 + -114a + 19a2 = 69 + -69 Combine like terms: 69 + -69 = 0 -69 + -114a + 19a2 = 0 Begin completing the square. Divide all terms by 19 the coefficient of the squared term: Divide each side by '19'. -3.631578947 + -6a + a2 = 0 Move the constant term to the right: Add '3.631578947' to each side of the equation. -3.631578947 + -6a + 3.631578947 + a2 = 0 + 3.631578947 Reorder the terms: -3.631578947 + 3.631578947 + -6a + a2 = 0 + 3.631578947 Combine like terms: -3.631578947 + 3.631578947 = 0.000000000 0.000000000 + -6a + a2 = 0 + 3.631578947 -6a + a2 = 0 + 3.631578947 Combine like terms: 0 + 3.631578947 = 3.631578947 -6a + a2 = 3.631578947 The a term is -6a. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6a + 9 + a2 = 3.631578947 + 9 Reorder the terms: 9 + -6a + a2 = 3.631578947 + 9 Combine like terms: 3.631578947 + 9 = 12.631578947 9 + -6a + a2 = 12.631578947 Factor a perfect square on the left side: (a + -3)(a + -3) = 12.631578947 Calculate the square root of the right side: 3.554093267 Break this problem into two subproblems by setting (a + -3) equal to 3.554093267 and -3.554093267.Subproblem 1
a + -3 = 3.554093267 Simplifying a + -3 = 3.554093267 Reorder the terms: -3 + a = 3.554093267 Solving -3 + a = 3.554093267 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + a = 3.554093267 + 3 Combine like terms: -3 + 3 = 0 0 + a = 3.554093267 + 3 a = 3.554093267 + 3 Combine like terms: 3.554093267 + 3 = 6.554093267 a = 6.554093267 Simplifying a = 6.554093267Subproblem 2
a + -3 = -3.554093267 Simplifying a + -3 = -3.554093267 Reorder the terms: -3 + a = -3.554093267 Solving -3 + a = -3.554093267 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + a = -3.554093267 + 3 Combine like terms: -3 + 3 = 0 0 + a = -3.554093267 + 3 a = -3.554093267 + 3 Combine like terms: -3.554093267 + 3 = -0.554093267 a = -0.554093267 Simplifying a = -0.554093267Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.554093267, -0.554093267}
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