135=0.006a^2-0.02a+120

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Solution for 135=0.006a^2-0.02a+120 equation:


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 = 51.694436733

Subproblem 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 = -48.361103399

Solution

The solution to the problem is based on the solutions from the subproblems. a = {51.694436733, -48.361103399}

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