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Simplifying 8a2 = 15 + -8a Solving 8a2 = 15 + -8a Solving for variable 'a'. Reorder the terms: -15 + 8a + 8a2 = 15 + -8a + -15 + 8a Reorder the terms: -15 + 8a + 8a2 = 15 + -15 + -8a + 8a Combine like terms: 15 + -15 = 0 -15 + 8a + 8a2 = 0 + -8a + 8a -15 + 8a + 8a2 = -8a + 8a Combine like terms: -8a + 8a = 0 -15 + 8a + 8a2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.875 + a + a2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + a + 1.875 + a2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + a + a2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + a + a2 = 0 + 1.875 a + a2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 a + a2 = 1.875 The a term is a. 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. + 0.25 + a2 = 1.875 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = 1.875 + 0.25 Combine like terms: 1.875 + 0.25 = 2.125 1.25 + a2 = 2.125 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 2.125 Calculate the square root of the right side: 1.457737974 Break this problem into two subproblems by setting (a + 0.5) equal to 1.457737974 and -1.457737974.Subproblem 1
a + 0.5 = 1.457737974 Simplifying a + 0.5 = 1.457737974 Reorder the terms: 0.5 + a = 1.457737974 Solving 0.5 + a = 1.457737974 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 = 1.457737974 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.457737974 + -0.5 a = 1.457737974 + -0.5 Combine like terms: 1.457737974 + -0.5 = 0.957737974 a = 0.957737974 Simplifying a = 0.957737974Subproblem 2
a + 0.5 = -1.457737974 Simplifying a + 0.5 = -1.457737974 Reorder the terms: 0.5 + a = -1.457737974 Solving 0.5 + a = -1.457737974 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 = -1.457737974 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.457737974 + -0.5 a = -1.457737974 + -0.5 Combine like terms: -1.457737974 + -0.5 = -1.957737974 a = -1.957737974 Simplifying a = -1.957737974Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.957737974, -1.957737974}
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