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Simplifying 30a + a2 = (60 + -2a) + -20 Remove parenthesis around (60 + -2a) 30a + a2 = 60 + -2a + -20 Reorder the terms: 30a + a2 = 60 + -20 + -2a Combine like terms: 60 + -20 = 40 30a + a2 = 40 + -2a Solving 30a + a2 = 40 + -2a Solving for variable 'a'. Reorder the terms: -40 + 30a + 2a + a2 = 40 + -2a + -40 + 2a Combine like terms: 30a + 2a = 32a -40 + 32a + a2 = 40 + -2a + -40 + 2a Reorder the terms: -40 + 32a + a2 = 40 + -40 + -2a + 2a Combine like terms: 40 + -40 = 0 -40 + 32a + a2 = 0 + -2a + 2a -40 + 32a + a2 = -2a + 2a Combine like terms: -2a + 2a = 0 -40 + 32a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + 32a + 40 + a2 = 0 + 40 Reorder the terms: -40 + 40 + 32a + a2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 32a + a2 = 0 + 40 32a + a2 = 0 + 40 Combine like terms: 0 + 40 = 40 32a + a2 = 40 The a term is 32a. Take half its coefficient (16). Square it (256) and add it to both sides. Add '256' to each side of the equation. 32a + 256 + a2 = 40 + 256 Reorder the terms: 256 + 32a + a2 = 40 + 256 Combine like terms: 40 + 256 = 296 256 + 32a + a2 = 296 Factor a perfect square on the left side: (a + 16)(a + 16) = 296 Calculate the square root of the right side: 17.204650534 Break this problem into two subproblems by setting (a + 16) equal to 17.204650534 and -17.204650534.Subproblem 1
a + 16 = 17.204650534 Simplifying a + 16 = 17.204650534 Reorder the terms: 16 + a = 17.204650534 Solving 16 + a = 17.204650534 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-16' to each side of the equation. 16 + -16 + a = 17.204650534 + -16 Combine like terms: 16 + -16 = 0 0 + a = 17.204650534 + -16 a = 17.204650534 + -16 Combine like terms: 17.204650534 + -16 = 1.204650534 a = 1.204650534 Simplifying a = 1.204650534Subproblem 2
a + 16 = -17.204650534 Simplifying a + 16 = -17.204650534 Reorder the terms: 16 + a = -17.204650534 Solving 16 + a = -17.204650534 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-16' to each side of the equation. 16 + -16 + a = -17.204650534 + -16 Combine like terms: 16 + -16 = 0 0 + a = -17.204650534 + -16 a = -17.204650534 + -16 Combine like terms: -17.204650534 + -16 = -33.204650534 a = -33.204650534 Simplifying a = -33.204650534Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.204650534, -33.204650534}
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