11a^2-32a+7=20

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Solution for 11a^2-32a+7=20 equation:


Simplifying
11a2 + -32a + 7 = 20

Reorder the terms:
7 + -32a + 11a2 = 20

Solving
7 + -32a + 11a2 = 20

Solving for variable 'a'.

Reorder the terms:
7 + -20 + -32a + 11a2 = 20 + -20

Combine like terms: 7 + -20 = -13
-13 + -32a + 11a2 = 20 + -20

Combine like terms: 20 + -20 = 0
-13 + -32a + 11a2 = 0

Begin completing the square.  Divide all terms by
11 the coefficient of the squared term: 

Divide each side by '11'.
-1.181818182 + -2.909090909a + a2 = 0

Move the constant term to the right:

Add '1.181818182' to each side of the equation.
-1.181818182 + -2.909090909a + 1.181818182 + a2 = 0 + 1.181818182

Reorder the terms:
-1.181818182 + 1.181818182 + -2.909090909a + a2 = 0 + 1.181818182

Combine like terms: -1.181818182 + 1.181818182 = 0.000000000
0.000000000 + -2.909090909a + a2 = 0 + 1.181818182
-2.909090909a + a2 = 0 + 1.181818182

Combine like terms: 0 + 1.181818182 = 1.181818182
-2.909090909a + a2 = 1.181818182

The a term is -2.909090909a.  Take half its coefficient (-1.454545455).
Square it (2.115702481) and add it to both sides.

Add '2.115702481' to each side of the equation.
-2.909090909a + 2.115702481 + a2 = 1.181818182 + 2.115702481

Reorder the terms:
2.115702481 + -2.909090909a + a2 = 1.181818182 + 2.115702481

Combine like terms: 1.181818182 + 2.115702481 = 3.297520663
2.115702481 + -2.909090909a + a2 = 3.297520663

Factor a perfect square on the left side:
(a + -1.454545455)(a + -1.454545455) = 3.297520663

Calculate the square root of the right side: 1.815907669

Break this problem into two subproblems by setting 
(a + -1.454545455) equal to 1.815907669 and -1.815907669.

Subproblem 1

a + -1.454545455 = 1.815907669 Simplifying a + -1.454545455 = 1.815907669 Reorder the terms: -1.454545455 + a = 1.815907669 Solving -1.454545455 + a = 1.815907669 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.454545455' to each side of the equation. -1.454545455 + 1.454545455 + a = 1.815907669 + 1.454545455 Combine like terms: -1.454545455 + 1.454545455 = 0.000000000 0.000000000 + a = 1.815907669 + 1.454545455 a = 1.815907669 + 1.454545455 Combine like terms: 1.815907669 + 1.454545455 = 3.270453124 a = 3.270453124 Simplifying a = 3.270453124

Subproblem 2

a + -1.454545455 = -1.815907669 Simplifying a + -1.454545455 = -1.815907669 Reorder the terms: -1.454545455 + a = -1.815907669 Solving -1.454545455 + a = -1.815907669 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.454545455' to each side of the equation. -1.454545455 + 1.454545455 + a = -1.815907669 + 1.454545455 Combine like terms: -1.454545455 + 1.454545455 = 0.000000000 0.000000000 + a = -1.815907669 + 1.454545455 a = -1.815907669 + 1.454545455 Combine like terms: -1.815907669 + 1.454545455 = -0.361362214 a = -0.361362214 Simplifying a = -0.361362214

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.270453124, -0.361362214}

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