3a^2=11a+3

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Solution for 3a^2=11a+3 equation:


Simplifying
3a2 = 11a + 3

Reorder the terms:
3a2 = 3 + 11a

Solving
3a2 = 3 + 11a

Solving for variable 'a'.

Reorder the terms:
-3 + -11a + 3a2 = 3 + 11a + -3 + -11a

Reorder the terms:
-3 + -11a + 3a2 = 3 + -3 + 11a + -11a

Combine like terms: 3 + -3 = 0
-3 + -11a + 3a2 = 0 + 11a + -11a
-3 + -11a + 3a2 = 11a + -11a

Combine like terms: 11a + -11a = 0
-3 + -11a + 3a2 = 0

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

Divide each side by '3'.
-1 + -3.666666667a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3.666666667a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + -3.666666667a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3.666666667a + a2 = 0 + 1
-3.666666667a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3.666666667a + a2 = 1

The a term is -3.666666667a.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667a + 3.361111114 + a2 = 1 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667a + a2 = 1 + 3.361111114

Combine like terms: 1 + 3.361111114 = 4.361111114
3.361111114 + -3.666666667a + a2 = 4.361111114

Factor a perfect square on the left side:
(a + -1.833333334)(a + -1.833333334) = 4.361111114

Calculate the square root of the right side: 2.088327348

Break this problem into two subproblems by setting 
(a + -1.833333334) equal to 2.088327348 and -2.088327348.

Subproblem 1

a + -1.833333334 = 2.088327348 Simplifying a + -1.833333334 = 2.088327348 Reorder the terms: -1.833333334 + a = 2.088327348 Solving -1.833333334 + a = 2.088327348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + a = 2.088327348 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = 2.088327348 + 1.833333334 a = 2.088327348 + 1.833333334 Combine like terms: 2.088327348 + 1.833333334 = 3.921660682 a = 3.921660682 Simplifying a = 3.921660682

Subproblem 2

a + -1.833333334 = -2.088327348 Simplifying a + -1.833333334 = -2.088327348 Reorder the terms: -1.833333334 + a = -2.088327348 Solving -1.833333334 + a = -2.088327348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + a = -2.088327348 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = -2.088327348 + 1.833333334 a = -2.088327348 + 1.833333334 Combine like terms: -2.088327348 + 1.833333334 = -0.254994014 a = -0.254994014 Simplifying a = -0.254994014

Solution

The solution to the problem is based on the solutions from the subproblems. a = {3.921660682, -0.254994014}

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