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Simplifying 135 = 0.006a2 + -0.02a + 120 Reorder the terms: 135 = 120 + -0.02a + 0.006a2 Solving 135 = 120 + -0.02a + 0.006a2 Solving for variable 'a'. Combine like terms: 135 + -120 = 15 15 + 0.02a + -0.006a2 = 120 + -0.02a + 0.006a2 + -120 + 0.02a + -0.006a2 Reorder the terms: 15 + 0.02a + -0.006a2 = 120 + -120 + -0.02a + 0.02a + 0.006a2 + -0.006a2 Combine like terms: 120 + -120 = 0 15 + 0.02a + -0.006a2 = 0 + -0.02a + 0.02a + 0.006a2 + -0.006a2 15 + 0.02a + -0.006a2 = -0.02a + 0.02a + 0.006a2 + -0.006a2 Combine like terms: -0.02a + 0.02a = 0.00 15 + 0.02a + -0.006a2 = 0.00 + 0.006a2 + -0.006a2 15 + 0.02a + -0.006a2 = 0.006a2 + -0.006a2 Combine like terms: 0.006a2 + -0.006a2 = 0.000 15 + 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'. -2500 + -3.333333333a + a2 = 0 Move the constant term to the right: Add '2500' to each side of the equation. -2500 + -3.333333333a + 2500 + a2 = 0 + 2500 Reorder the terms: -2500 + 2500 + -3.333333333a + a2 = 0 + 2500 Combine like terms: -2500 + 2500 = 0 0 + -3.333333333a + a2 = 0 + 2500 -3.333333333a + a2 = 0 + 2500 Combine like terms: 0 + 2500 = 2500 -3.333333333a + a2 = 2500 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 = 2500 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333a + a2 = 2500 + 2.777777779 Combine like terms: 2500 + 2.777777779 = 2502.777777779 2.777777779 + -3.333333333a + a2 = 2502.777777779 Factor a perfect square on the left side: (a + -1.666666667)(a + -1.666666667) = 2502.777777779 Calculate the square root of the right side: 50.027770066 Break this problem into two subproblems by setting (a + -1.666666667) equal to 50.027770066 and -50.027770066.Subproblem 1
a + -1.666666667 = 50.027770066 Simplifying a + -1.666666667 = 50.027770066 Reorder the terms: -1.666666667 + a = 50.027770066 Solving -1.666666667 + a = 50.027770066 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 = 50.027770066 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = 50.027770066 + 1.666666667 a = 50.027770066 + 1.666666667 Combine like terms: 50.027770066 + 1.666666667 = 51.694436733 a = 51.694436733 Simplifying a = 51.694436733Subproblem 2
a + -1.666666667 = -50.027770066 Simplifying a + -1.666666667 = -50.027770066 Reorder the terms: -1.666666667 + a = -50.027770066 Solving -1.666666667 + a = -50.027770066 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 = -50.027770066 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + a = -50.027770066 + 1.666666667 a = -50.027770066 + 1.666666667 Combine like terms: -50.027770066 + 1.666666667 = -48.361103399 a = -48.361103399 Simplifying a = -48.361103399Solution
The solution to the problem is based on the solutions from the subproblems. a = {51.694436733, -48.361103399}
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