9a^2=12a+1

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Solution for 9a^2=12a+1 equation:


Simplifying
9a2 = 12a + 1

Reorder the terms:
9a2 = 1 + 12a

Solving
9a2 = 1 + 12a

Solving for variable 'a'.

Reorder the terms:
-1 + -12a + 9a2 = 1 + 12a + -1 + -12a

Reorder the terms:
-1 + -12a + 9a2 = 1 + -1 + 12a + -12a

Combine like terms: 1 + -1 = 0
-1 + -12a + 9a2 = 0 + 12a + -12a
-1 + -12a + 9a2 = 12a + -12a

Combine like terms: 12a + -12a = 0
-1 + -12a + 9a2 = 0

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

Divide each side by '9'.
-0.1111111111 + -1.333333333a + a2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -1.333333333a + 0.1111111111 + a2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -1.333333333a + a2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -1.333333333a + a2 = 0 + 0.1111111111
-1.333333333a + a2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-1.333333333a + a2 = 0.1111111111

The a term is -1.333333333a.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333a + 0.4444444442 + a2 = 0.1111111111 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333a + a2 = 0.1111111111 + 0.4444444442

Combine like terms: 0.1111111111 + 0.4444444442 = 0.5555555553
0.4444444442 + -1.333333333a + a2 = 0.5555555553

Factor a perfect square on the left side:
(a + -0.6666666665)(a + -0.6666666665) = 0.5555555553

Calculate the square root of the right side: 0.745355992

Break this problem into two subproblems by setting 
(a + -0.6666666665) equal to 0.745355992 and -0.745355992.

Subproblem 1

a + -0.6666666665 = 0.745355992 Simplifying a + -0.6666666665 = 0.745355992 Reorder the terms: -0.6666666665 + a = 0.745355992 Solving -0.6666666665 + a = 0.745355992 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = 0.745355992 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = 0.745355992 + 0.6666666665 a = 0.745355992 + 0.6666666665 Combine like terms: 0.745355992 + 0.6666666665 = 1.4120226585 a = 1.4120226585 Simplifying a = 1.4120226585

Subproblem 2

a + -0.6666666665 = -0.745355992 Simplifying a + -0.6666666665 = -0.745355992 Reorder the terms: -0.6666666665 + a = -0.745355992 Solving -0.6666666665 + a = -0.745355992 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = -0.745355992 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = -0.745355992 + 0.6666666665 a = -0.745355992 + 0.6666666665 Combine like terms: -0.745355992 + 0.6666666665 = -0.0786893255 a = -0.0786893255 Simplifying a = -0.0786893255

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.4120226585, -0.0786893255}

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