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Simplifying 32a2 + -19a = 63 Reorder the terms: -19a + 32a2 = 63 Solving -19a + 32a2 = 63 Solving for variable 'a'. Reorder the terms: -63 + -19a + 32a2 = 63 + -63 Combine like terms: 63 + -63 = 0 -63 + -19a + 32a2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -1.96875 + -0.59375a + a2 = 0 Move the constant term to the right: Add '1.96875' to each side of the equation. -1.96875 + -0.59375a + 1.96875 + a2 = 0 + 1.96875 Reorder the terms: -1.96875 + 1.96875 + -0.59375a + a2 = 0 + 1.96875 Combine like terms: -1.96875 + 1.96875 = 0.00000 0.00000 + -0.59375a + a2 = 0 + 1.96875 -0.59375a + a2 = 0 + 1.96875 Combine like terms: 0 + 1.96875 = 1.96875 -0.59375a + a2 = 1.96875 The a term is -0.59375a. Take half its coefficient (-0.296875). Square it (0.08813476563) and add it to both sides. Add '0.08813476563' to each side of the equation. -0.59375a + 0.08813476563 + a2 = 1.96875 + 0.08813476563 Reorder the terms: 0.08813476563 + -0.59375a + a2 = 1.96875 + 0.08813476563 Combine like terms: 1.96875 + 0.08813476563 = 2.05688476563 0.08813476563 + -0.59375a + a2 = 2.05688476563 Factor a perfect square on the left side: (a + -0.296875)(a + -0.296875) = 2.05688476563 Calculate the square root of the right side: 1.434184356 Break this problem into two subproblems by setting (a + -0.296875) equal to 1.434184356 and -1.434184356.Subproblem 1
a + -0.296875 = 1.434184356 Simplifying a + -0.296875 = 1.434184356 Reorder the terms: -0.296875 + a = 1.434184356 Solving -0.296875 + a = 1.434184356 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.296875' to each side of the equation. -0.296875 + 0.296875 + a = 1.434184356 + 0.296875 Combine like terms: -0.296875 + 0.296875 = 0.000000 0.000000 + a = 1.434184356 + 0.296875 a = 1.434184356 + 0.296875 Combine like terms: 1.434184356 + 0.296875 = 1.731059356 a = 1.731059356 Simplifying a = 1.731059356Subproblem 2
a + -0.296875 = -1.434184356 Simplifying a + -0.296875 = -1.434184356 Reorder the terms: -0.296875 + a = -1.434184356 Solving -0.296875 + a = -1.434184356 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.296875' to each side of the equation. -0.296875 + 0.296875 + a = -1.434184356 + 0.296875 Combine like terms: -0.296875 + 0.296875 = 0.000000 0.000000 + a = -1.434184356 + 0.296875 a = -1.434184356 + 0.296875 Combine like terms: -1.434184356 + 0.296875 = -1.137309356 a = -1.137309356 Simplifying a = -1.137309356Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.731059356, -1.137309356}
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