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Simplifying 138 = 0.006A2 + -0.02A + 120 Reorder the terms: 138 = 120 + -0.02A + 0.006A2 Solving 138 = 120 + -0.02A + 0.006A2 Solving for variable 'A'. Combine like terms: 138 + -120 = 18 18 + 0.02A + -0.006A2 = 120 + -0.02A + 0.006A2 + -120 + 0.02A + -0.006A2 Reorder the terms: 18 + 0.02A + -0.006A2 = 120 + -120 + -0.02A + 0.02A + 0.006A2 + -0.006A2 Combine like terms: 120 + -120 = 0 18 + 0.02A + -0.006A2 = 0 + -0.02A + 0.02A + 0.006A2 + -0.006A2 18 + 0.02A + -0.006A2 = -0.02A + 0.02A + 0.006A2 + -0.006A2 Combine like terms: -0.02A + 0.02A = 0.00 18 + 0.02A + -0.006A2 = 0.00 + 0.006A2 + -0.006A2 18 + 0.02A + -0.006A2 = 0.006A2 + -0.006A2 Combine like terms: 0.006A2 + -0.006A2 = 0.000 18 + 0.02A + -0.006A2 = 0.000 Begin completing the square. Divide all terms by -0.006 the coefficient of the squared term: Divide each side by '-0.006'. -3000 + -3.333333333A + A2 = 0 Move the constant term to the right: Add '3000' to each side of the equation. -3000 + -3.333333333A + 3000 + A2 = 0 + 3000 Reorder the terms: -3000 + 3000 + -3.333333333A + A2 = 0 + 3000 Combine like terms: -3000 + 3000 = 0 0 + -3.333333333A + A2 = 0 + 3000 -3.333333333A + A2 = 0 + 3000 Combine like terms: 0 + 3000 = 3000 -3.333333333A + A2 = 3000 The A term is -3.333333333A. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333A + 2.777777779 + A2 = 3000 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333A + A2 = 3000 + 2.777777779 Combine like terms: 3000 + 2.777777779 = 3002.777777779 2.777777779 + -3.333333333A + A2 = 3002.777777779 Factor a perfect square on the left side: (A + -1.666666667)(A + -1.666666667) = 3002.777777779 Calculate the square root of the right side: 54.797607409 Break this problem into two subproblems by setting (A + -1.666666667) equal to 54.797607409 and -54.797607409.Subproblem 1
A + -1.666666667 = 54.797607409 Simplifying A + -1.666666667 = 54.797607409 Reorder the terms: -1.666666667 + A = 54.797607409 Solving -1.666666667 + A = 54.797607409 Solving for variable 'A'. Move all terms containing A to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + A = 54.797607409 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + A = 54.797607409 + 1.666666667 A = 54.797607409 + 1.666666667 Combine like terms: 54.797607409 + 1.666666667 = 56.464274076 A = 56.464274076 Simplifying A = 56.464274076Subproblem 2
A + -1.666666667 = -54.797607409 Simplifying A + -1.666666667 = -54.797607409 Reorder the terms: -1.666666667 + A = -54.797607409 Solving -1.666666667 + A = -54.797607409 Solving for variable 'A'. Move all terms containing A to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + A = -54.797607409 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + A = -54.797607409 + 1.666666667 A = -54.797607409 + 1.666666667 Combine like terms: -54.797607409 + 1.666666667 = -53.130940742 A = -53.130940742 Simplifying A = -53.130940742Solution
The solution to the problem is based on the solutions from the subproblems. A = {56.464274076, -53.130940742}
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