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Simplifying 2a2 + 10a + -27 = -2 Reorder the terms: -27 + 10a + 2a2 = -2 Solving -27 + 10a + 2a2 = -2 Solving for variable 'a'. Reorder the terms: -27 + 2 + 10a + 2a2 = -2 + 2 Combine like terms: -27 + 2 = -25 -25 + 10a + 2a2 = -2 + 2 Combine like terms: -2 + 2 = 0 -25 + 10a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -12.5 + 5a + a2 = 0 Move the constant term to the right: Add '12.5' to each side of the equation. -12.5 + 5a + 12.5 + a2 = 0 + 12.5 Reorder the terms: -12.5 + 12.5 + 5a + a2 = 0 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + 5a + a2 = 0 + 12.5 5a + a2 = 0 + 12.5 Combine like terms: 0 + 12.5 = 12.5 5a + a2 = 12.5 The a term is 5a. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5a + 6.25 + a2 = 12.5 + 6.25 Reorder the terms: 6.25 + 5a + a2 = 12.5 + 6.25 Combine like terms: 12.5 + 6.25 = 18.75 6.25 + 5a + a2 = 18.75 Factor a perfect square on the left side: (a + 2.5)(a + 2.5) = 18.75 Calculate the square root of the right side: 4.330127019 Break this problem into two subproblems by setting (a + 2.5) equal to 4.330127019 and -4.330127019.Subproblem 1
a + 2.5 = 4.330127019 Simplifying a + 2.5 = 4.330127019 Reorder the terms: 2.5 + a = 4.330127019 Solving 2.5 + a = 4.330127019 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = 4.330127019 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = 4.330127019 + -2.5 a = 4.330127019 + -2.5 Combine like terms: 4.330127019 + -2.5 = 1.830127019 a = 1.830127019 Simplifying a = 1.830127019Subproblem 2
a + 2.5 = -4.330127019 Simplifying a + 2.5 = -4.330127019 Reorder the terms: 2.5 + a = -4.330127019 Solving 2.5 + a = -4.330127019 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = -4.330127019 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = -4.330127019 + -2.5 a = -4.330127019 + -2.5 Combine like terms: -4.330127019 + -2.5 = -6.830127019 a = -6.830127019 Simplifying a = -6.830127019Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.830127019, -6.830127019}
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