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Simplifying 15a2 + 12a + -65 = 0 Reorder the terms: -65 + 12a + 15a2 = 0 Solving -65 + 12a + 15a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -4.333333333 + 0.8a + a2 = 0 Move the constant term to the right: Add '4.333333333' to each side of the equation. -4.333333333 + 0.8a + 4.333333333 + a2 = 0 + 4.333333333 Reorder the terms: -4.333333333 + 4.333333333 + 0.8a + a2 = 0 + 4.333333333 Combine like terms: -4.333333333 + 4.333333333 = 0.000000000 0.000000000 + 0.8a + a2 = 0 + 4.333333333 0.8a + a2 = 0 + 4.333333333 Combine like terms: 0 + 4.333333333 = 4.333333333 0.8a + a2 = 4.333333333 The a term is 0.8a. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8a + 0.16 + a2 = 4.333333333 + 0.16 Reorder the terms: 0.16 + 0.8a + a2 = 4.333333333 + 0.16 Combine like terms: 4.333333333 + 0.16 = 4.493333333 0.16 + 0.8a + a2 = 4.493333333 Factor a perfect square on the left side: (a + 0.4)(a + 0.4) = 4.493333333 Calculate the square root of the right side: 2.119748413 Break this problem into two subproblems by setting (a + 0.4) equal to 2.119748413 and -2.119748413.Subproblem 1
a + 0.4 = 2.119748413 Simplifying a + 0.4 = 2.119748413 Reorder the terms: 0.4 + a = 2.119748413 Solving 0.4 + a = 2.119748413 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + a = 2.119748413 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + a = 2.119748413 + -0.4 a = 2.119748413 + -0.4 Combine like terms: 2.119748413 + -0.4 = 1.719748413 a = 1.719748413 Simplifying a = 1.719748413Subproblem 2
a + 0.4 = -2.119748413 Simplifying a + 0.4 = -2.119748413 Reorder the terms: 0.4 + a = -2.119748413 Solving 0.4 + a = -2.119748413 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + a = -2.119748413 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + a = -2.119748413 + -0.4 a = -2.119748413 + -0.4 Combine like terms: -2.119748413 + -0.4 = -2.519748413 a = -2.519748413 Simplifying a = -2.519748413Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.719748413, -2.519748413}
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