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Simplifying a(5a + -3) = 2(a2 + 5) Reorder the terms: a(-3 + 5a) = 2(a2 + 5) (-3 * a + 5a * a) = 2(a2 + 5) (-3a + 5a2) = 2(a2 + 5) Reorder the terms: -3a + 5a2 = 2(5 + a2) -3a + 5a2 = (5 * 2 + a2 * 2) -3a + 5a2 = (10 + 2a2) Solving -3a + 5a2 = 10 + 2a2 Solving for variable 'a'. Reorder the terms: -10 + -3a + 5a2 + -2a2 = 10 + 2a2 + -10 + -2a2 Combine like terms: 5a2 + -2a2 = 3a2 -10 + -3a + 3a2 = 10 + 2a2 + -10 + -2a2 Reorder the terms: -10 + -3a + 3a2 = 10 + -10 + 2a2 + -2a2 Combine like terms: 10 + -10 = 0 -10 + -3a + 3a2 = 0 + 2a2 + -2a2 -10 + -3a + 3a2 = 2a2 + -2a2 Combine like terms: 2a2 + -2a2 = 0 -10 + -3a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3.333333333 + -1a + a2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + -1a + 3.333333333 + a2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + -1a + a2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + -1a + a2 = 0 + 3.333333333 -1a + a2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 -1a + a2 = 3.333333333 The a term is -1a. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1a + 0.25 + a2 = 3.333333333 + 0.25 Reorder the terms: 0.25 + -1a + a2 = 3.333333333 + 0.25 Combine like terms: 3.333333333 + 0.25 = 3.583333333 0.25 + -1a + a2 = 3.583333333 Factor a perfect square on the left side: (a + -0.5)(a + -0.5) = 3.583333333 Calculate the square root of the right side: 1.892969449 Break this problem into two subproblems by setting (a + -0.5) equal to 1.892969449 and -1.892969449.Subproblem 1
a + -0.5 = 1.892969449 Simplifying a + -0.5 = 1.892969449 Reorder the terms: -0.5 + a = 1.892969449 Solving -0.5 + a = 1.892969449 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = 1.892969449 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 1.892969449 + 0.5 a = 1.892969449 + 0.5 Combine like terms: 1.892969449 + 0.5 = 2.392969449 a = 2.392969449 Simplifying a = 2.392969449Subproblem 2
a + -0.5 = -1.892969449 Simplifying a + -0.5 = -1.892969449 Reorder the terms: -0.5 + a = -1.892969449 Solving -0.5 + a = -1.892969449 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = -1.892969449 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -1.892969449 + 0.5 a = -1.892969449 + 0.5 Combine like terms: -1.892969449 + 0.5 = -1.392969449 a = -1.392969449 Simplifying a = -1.392969449Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.392969449, -1.392969449}
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