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Simplifying [[(9a2 + 18a + 5) + (12a2 + 6a + 7)] + -1(23a2 + -10a + -1)] = 0 Reorder the terms: [[(5 + 18a + 9a2) + (12a2 + 6a + 7)] + -1(23a2 + -10a + -1)] = 0 Remove parenthesis around (5 + 18a + 9a2) [[5 + 18a + 9a2 + (12a2 + 6a + 7)] + -1(23a2 + -10a + -1)] = 0 Reorder the terms: [[5 + 18a + 9a2 + (7 + 6a + 12a2)] + -1(23a2 + -10a + -1)] = 0 Remove parenthesis around (7 + 6a + 12a2) [[5 + 18a + 9a2 + 7 + 6a + 12a2] + -1(23a2 + -10a + -1)] = 0 Reorder the terms: [[5 + 7 + 18a + 6a + 9a2 + 12a2] + -1(23a2 + -10a + -1)] = 0 Combine like terms: 5 + 7 = 12 [[12 + 18a + 6a + 9a2 + 12a2] + -1(23a2 + -10a + -1)] = 0 Combine like terms: 18a + 6a = 24a [[12 + 24a + 9a2 + 12a2] + -1(23a2 + -10a + -1)] = 0 Combine like terms: 9a2 + 12a2 = 21a2 [[12 + 24a + 21a2] + -1(23a2 + -10a + -1)] = 0 Remove brackets around [12 + 24a + 21a2] [12 + 24a + 21a2 + -1(23a2 + -10a + -1)] = 0 Reorder the terms: [12 + 24a + 21a2 + -1(-1 + -10a + 23a2)] = 0 [12 + 24a + 21a2 + (-1 * -1 + -10a * -1 + 23a2 * -1)] = 0 [12 + 24a + 21a2 + (1 + 10a + -23a2)] = 0 Reorder the terms: [12 + 1 + 24a + 10a + 21a2 + -23a2] = 0 Combine like terms: 12 + 1 = 13 [13 + 24a + 10a + 21a2 + -23a2] = 0 Combine like terms: 24a + 10a = 34a [13 + 34a + 21a2 + -23a2] = 0 Combine like terms: 21a2 + -23a2 = -2a2 [13 + 34a + -2a2] = 0 Remove brackets around [13 + 34a + -2a2] 13 + 34a + -2a2 = 0 Solving 13 + 34a + -2a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -6.5 + -17a + a2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + -17a + 6.5 + a2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + -17a + a2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + -17a + a2 = 0 + 6.5 -17a + a2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 -17a + a2 = 6.5 The a term is -17a. Take half its coefficient (-8.5). Square it (72.25) and add it to both sides. Add '72.25' to each side of the equation. -17a + 72.25 + a2 = 6.5 + 72.25 Reorder the terms: 72.25 + -17a + a2 = 6.5 + 72.25 Combine like terms: 6.5 + 72.25 = 78.75 72.25 + -17a + a2 = 78.75 Factor a perfect square on the left side: (a + -8.5)(a + -8.5) = 78.75 Calculate the square root of the right side: 8.874119675 Break this problem into two subproblems by setting (a + -8.5) equal to 8.874119675 and -8.874119675.Subproblem 1
a + -8.5 = 8.874119675 Simplifying a + -8.5 = 8.874119675 Reorder the terms: -8.5 + a = 8.874119675 Solving -8.5 + a = 8.874119675 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + a = 8.874119675 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + a = 8.874119675 + 8.5 a = 8.874119675 + 8.5 Combine like terms: 8.874119675 + 8.5 = 17.374119675 a = 17.374119675 Simplifying a = 17.374119675Subproblem 2
a + -8.5 = -8.874119675 Simplifying a + -8.5 = -8.874119675 Reorder the terms: -8.5 + a = -8.874119675 Solving -8.5 + a = -8.874119675 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + a = -8.874119675 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + a = -8.874119675 + 8.5 a = -8.874119675 + 8.5 Combine like terms: -8.874119675 + 8.5 = -0.374119675 a = -0.374119675 Simplifying a = -0.374119675Solution
The solution to the problem is based on the solutions from the subproblems. a = {17.374119675, -0.374119675}
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