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Simplifying 5a2 + -45a = -2a2 + 18 Reorder the terms: -45a + 5a2 = -2a2 + 18 Reorder the terms: -45a + 5a2 = 18 + -2a2 Solving -45a + 5a2 = 18 + -2a2 Solving for variable 'a'. Reorder the terms: -18 + -45a + 5a2 + 2a2 = 18 + -2a2 + -18 + 2a2 Combine like terms: 5a2 + 2a2 = 7a2 -18 + -45a + 7a2 = 18 + -2a2 + -18 + 2a2 Reorder the terms: -18 + -45a + 7a2 = 18 + -18 + -2a2 + 2a2 Combine like terms: 18 + -18 = 0 -18 + -45a + 7a2 = 0 + -2a2 + 2a2 -18 + -45a + 7a2 = -2a2 + 2a2 Combine like terms: -2a2 + 2a2 = 0 -18 + -45a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.571428571 + -6.428571429a + a2 = 0 Move the constant term to the right: Add '2.571428571' to each side of the equation. -2.571428571 + -6.428571429a + 2.571428571 + a2 = 0 + 2.571428571 Reorder the terms: -2.571428571 + 2.571428571 + -6.428571429a + a2 = 0 + 2.571428571 Combine like terms: -2.571428571 + 2.571428571 = 0.000000000 0.000000000 + -6.428571429a + a2 = 0 + 2.571428571 -6.428571429a + a2 = 0 + 2.571428571 Combine like terms: 0 + 2.571428571 = 2.571428571 -6.428571429a + a2 = 2.571428571 The a term is -6.428571429a. Take half its coefficient (-3.214285715). Square it (10.33163266) and add it to both sides. Add '10.33163266' to each side of the equation. -6.428571429a + 10.33163266 + a2 = 2.571428571 + 10.33163266 Reorder the terms: 10.33163266 + -6.428571429a + a2 = 2.571428571 + 10.33163266 Combine like terms: 2.571428571 + 10.33163266 = 12.903061231 10.33163266 + -6.428571429a + a2 = 12.903061231 Factor a perfect square on the left side: (a + -3.214285715)(a + -3.214285715) = 12.903061231 Calculate the square root of the right side: 3.592083133 Break this problem into two subproblems by setting (a + -3.214285715) equal to 3.592083133 and -3.592083133.Subproblem 1
a + -3.214285715 = 3.592083133 Simplifying a + -3.214285715 = 3.592083133 Reorder the terms: -3.214285715 + a = 3.592083133 Solving -3.214285715 + a = 3.592083133 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.214285715' to each side of the equation. -3.214285715 + 3.214285715 + a = 3.592083133 + 3.214285715 Combine like terms: -3.214285715 + 3.214285715 = 0.000000000 0.000000000 + a = 3.592083133 + 3.214285715 a = 3.592083133 + 3.214285715 Combine like terms: 3.592083133 + 3.214285715 = 6.806368848 a = 6.806368848 Simplifying a = 6.806368848Subproblem 2
a + -3.214285715 = -3.592083133 Simplifying a + -3.214285715 = -3.592083133 Reorder the terms: -3.214285715 + a = -3.592083133 Solving -3.214285715 + a = -3.592083133 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.214285715' to each side of the equation. -3.214285715 + 3.214285715 + a = -3.592083133 + 3.214285715 Combine like terms: -3.214285715 + 3.214285715 = 0.000000000 0.000000000 + a = -3.592083133 + 3.214285715 a = -3.592083133 + 3.214285715 Combine like terms: -3.592083133 + 3.214285715 = -0.377797418 a = -0.377797418 Simplifying a = -0.377797418Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.806368848, -0.377797418}
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