3a^2=2a+9

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Solution for 3a^2=2a+9 equation:


Simplifying
3a2 = 2a + 9

Reorder the terms:
3a2 = 9 + 2a

Solving
3a2 = 9 + 2a

Solving for variable 'a'.

Reorder the terms:
-9 + -2a + 3a2 = 9 + 2a + -9 + -2a

Reorder the terms:
-9 + -2a + 3a2 = 9 + -9 + 2a + -2a

Combine like terms: 9 + -9 = 0
-9 + -2a + 3a2 = 0 + 2a + -2a
-9 + -2a + 3a2 = 2a + -2a

Combine like terms: 2a + -2a = 0
-9 + -2a + 3a2 = 0

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

Divide each side by '3'.
-3 + -0.6666666667a + a2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.6666666667a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.6666666667a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.6666666667a + a2 = 0 + 3
-0.6666666667a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.6666666667a + a2 = 3

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 = 3 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667a + a2 = 3 + 0.1111111112

Combine like terms: 3 + 0.1111111112 = 3.1111111112
0.1111111112 + -0.6666666667a + a2 = 3.1111111112

Factor a perfect square on the left side:
(a + -0.3333333334)(a + -0.3333333334) = 3.1111111112

Calculate the square root of the right side: 1.763834207

Break this problem into two subproblems by setting 
(a + -0.3333333334) equal to 1.763834207 and -1.763834207.

Subproblem 1

a + -0.3333333334 = 1.763834207 Simplifying a + -0.3333333334 = 1.763834207 Reorder the terms: -0.3333333334 + a = 1.763834207 Solving -0.3333333334 + a = 1.763834207 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 = 1.763834207 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = 1.763834207 + 0.3333333334 a = 1.763834207 + 0.3333333334 Combine like terms: 1.763834207 + 0.3333333334 = 2.0971675404 a = 2.0971675404 Simplifying a = 2.0971675404

Subproblem 2

a + -0.3333333334 = -1.763834207 Simplifying a + -0.3333333334 = -1.763834207 Reorder the terms: -0.3333333334 + a = -1.763834207 Solving -0.3333333334 + a = -1.763834207 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 = -1.763834207 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = -1.763834207 + 0.3333333334 a = -1.763834207 + 0.3333333334 Combine like terms: -1.763834207 + 0.3333333334 = -1.4305008736 a = -1.4305008736 Simplifying a = -1.4305008736

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.0971675404, -1.4305008736}

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