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Simplifying 0 = a2 + 3a + -90 Reorder the terms: 0 = -90 + 3a + a2 Solving 0 = -90 + 3a + a2 Solving for variable 'a'. Combine like terms: 0 + 90 = 90 90 + -3a + -1a2 = -90 + 3a + a2 + 90 + -3a + -1a2 Reorder the terms: 90 + -3a + -1a2 = -90 + 90 + 3a + -3a + a2 + -1a2 Combine like terms: -90 + 90 = 0 90 + -3a + -1a2 = 0 + 3a + -3a + a2 + -1a2 90 + -3a + -1a2 = 3a + -3a + a2 + -1a2 Combine like terms: 3a + -3a = 0 90 + -3a + -1a2 = 0 + a2 + -1a2 90 + -3a + -1a2 = a2 + -1a2 Combine like terms: a2 + -1a2 = 0 90 + -3a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -90 + 3a + a2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + 3a + 90 + a2 = 0 + 90 Reorder the terms: -90 + 90 + 3a + a2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 3a + a2 = 0 + 90 3a + a2 = 0 + 90 Combine like terms: 0 + 90 = 90 3a + a2 = 90 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 90 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 90 + 2.25 Combine like terms: 90 + 2.25 = 92.25 2.25 + 3a + a2 = 92.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 92.25 Calculate the square root of the right side: 9.604686356 Break this problem into two subproblems by setting (a + 1.5) equal to 9.604686356 and -9.604686356.Subproblem 1
a + 1.5 = 9.604686356 Simplifying a + 1.5 = 9.604686356 Reorder the terms: 1.5 + a = 9.604686356 Solving 1.5 + a = 9.604686356 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 9.604686356 + -1.5 a = 9.604686356 + -1.5 Combine like terms: 9.604686356 + -1.5 = 8.104686356 a = 8.104686356 Simplifying a = 8.104686356Subproblem 2
a + 1.5 = -9.604686356 Simplifying a + 1.5 = -9.604686356 Reorder the terms: 1.5 + a = -9.604686356 Solving 1.5 + a = -9.604686356 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -9.604686356 + -1.5 a = -9.604686356 + -1.5 Combine like terms: -9.604686356 + -1.5 = -11.104686356 a = -11.104686356 Simplifying a = -11.104686356Solution
The solution to the problem is based on the solutions from the subproblems. a = {8.104686356, -11.104686356}
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