6x^2+11x=45

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Solution for 6x^2+11x=45 equation:


Simplifying
6x2 + 11x = 45

Reorder the terms:
11x + 6x2 = 45

Solving
11x + 6x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 11x + 6x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 11x + 6x2 = 0

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

Divide each side by '6'.
-7.5 + 1.833333333x + x2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 1.833333333x + 7.5 + x2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 1.833333333x + x2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 1.833333333x + x2 = 0 + 7.5
1.833333333x + x2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
1.833333333x + x2 = 7.5

The x term is 1.833333333x.  Take half its coefficient (0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
1.833333333x + 0.8402777775 + x2 = 7.5 + 0.8402777775

Reorder the terms:
0.8402777775 + 1.833333333x + x2 = 7.5 + 0.8402777775

Combine like terms: 7.5 + 0.8402777775 = 8.3402777775
0.8402777775 + 1.833333333x + x2 = 8.3402777775

Factor a perfect square on the left side:
(x + 0.9166666665)(x + 0.9166666665) = 8.3402777775

Calculate the square root of the right side: 2.887953908

Break this problem into two subproblems by setting 
(x + 0.9166666665) equal to 2.887953908 and -2.887953908.

Subproblem 1

x + 0.9166666665 = 2.887953908 Simplifying x + 0.9166666665 = 2.887953908 Reorder the terms: 0.9166666665 + x = 2.887953908 Solving 0.9166666665 + x = 2.887953908 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = 2.887953908 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = 2.887953908 + -0.9166666665 x = 2.887953908 + -0.9166666665 Combine like terms: 2.887953908 + -0.9166666665 = 1.9712872415 x = 1.9712872415 Simplifying x = 1.9712872415

Subproblem 2

x + 0.9166666665 = -2.887953908 Simplifying x + 0.9166666665 = -2.887953908 Reorder the terms: 0.9166666665 + x = -2.887953908 Solving 0.9166666665 + x = -2.887953908 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = -2.887953908 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = -2.887953908 + -0.9166666665 x = -2.887953908 + -0.9166666665 Combine like terms: -2.887953908 + -0.9166666665 = -3.8046205745 x = -3.8046205745 Simplifying x = -3.8046205745

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.9712872415, -3.8046205745}

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