9a^2-11=6a

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Solution for 9a^2-11=6a equation:


Simplifying
9a2 + -11 = 6a

Reorder the terms:
-11 + 9a2 = 6a

Solving
-11 + 9a2 = 6a

Solving for variable 'a'.

Reorder the terms:
-11 + -6a + 9a2 = 6a + -6a

Combine like terms: 6a + -6a = 0
-11 + -6a + 9a2 = 0

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

Divide each side by '9'.
-1.222222222 + -0.6666666667a + a2 = 0

Move the constant term to the right:

Add '1.222222222' to each side of the equation.
-1.222222222 + -0.6666666667a + 1.222222222 + a2 = 0 + 1.222222222

Reorder the terms:
-1.222222222 + 1.222222222 + -0.6666666667a + a2 = 0 + 1.222222222

Combine like terms: -1.222222222 + 1.222222222 = 0.000000000
0.000000000 + -0.6666666667a + a2 = 0 + 1.222222222
-0.6666666667a + a2 = 0 + 1.222222222

Combine like terms: 0 + 1.222222222 = 1.222222222
-0.6666666667a + a2 = 1.222222222

The a term is -0.6666666667a.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667a + 0.1111111112 + a2 = 1.222222222 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667a + a2 = 1.222222222 + 0.1111111112

Combine like terms: 1.222222222 + 0.1111111112 = 1.3333333332
0.1111111112 + -0.6666666667a + a2 = 1.3333333332

Factor a perfect square on the left side:
(a + -0.3333333334)(a + -0.3333333334) = 1.3333333332

Calculate the square root of the right side: 1.154700538

Break this problem into two subproblems by setting 
(a + -0.3333333334) equal to 1.154700538 and -1.154700538.

Subproblem 1

a + -0.3333333334 = 1.154700538 Simplifying a + -0.3333333334 = 1.154700538 Reorder the terms: -0.3333333334 + a = 1.154700538 Solving -0.3333333334 + a = 1.154700538 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = 1.154700538 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = 1.154700538 + 0.3333333334 a = 1.154700538 + 0.3333333334 Combine like terms: 1.154700538 + 0.3333333334 = 1.4880338714 a = 1.4880338714 Simplifying a = 1.4880338714

Subproblem 2

a + -0.3333333334 = -1.154700538 Simplifying a + -0.3333333334 = -1.154700538 Reorder the terms: -0.3333333334 + a = -1.154700538 Solving -0.3333333334 + a = -1.154700538 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = -1.154700538 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = -1.154700538 + 0.3333333334 a = -1.154700538 + 0.3333333334 Combine like terms: -1.154700538 + 0.3333333334 = -0.8213672046 a = -0.8213672046 Simplifying a = -0.8213672046

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.4880338714, -0.8213672046}

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