2L^2+9L=8

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Solution for 2L^2+9L=8 equation:


Simplifying
2L2 + 9L = 8

Reorder the terms:
9L + 2L2 = 8

Solving
9L + 2L2 = 8

Solving for variable 'L'.

Reorder the terms:
-8 + 9L + 2L2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 9L + 2L2 = 0

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

Divide each side by '2'.
-4 + 4.5L + L2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 4.5L + 4 + L2 = 0 + 4

Reorder the terms:
-4 + 4 + 4.5L + L2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 4.5L + L2 = 0 + 4
4.5L + L2 = 0 + 4

Combine like terms: 0 + 4 = 4
4.5L + L2 = 4

The L term is 4.5L.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5L + 5.0625 + L2 = 4 + 5.0625

Reorder the terms:
5.0625 + 4.5L + L2 = 4 + 5.0625

Combine like terms: 4 + 5.0625 = 9.0625
5.0625 + 4.5L + L2 = 9.0625

Factor a perfect square on the left side:
(L + 2.25)(L + 2.25) = 9.0625

Calculate the square root of the right side: 3.010398645

Break this problem into two subproblems by setting 
(L + 2.25) equal to 3.010398645 and -3.010398645.

Subproblem 1

L + 2.25 = 3.010398645 Simplifying L + 2.25 = 3.010398645 Reorder the terms: 2.25 + L = 3.010398645 Solving 2.25 + L = 3.010398645 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + L = 3.010398645 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + L = 3.010398645 + -2.25 L = 3.010398645 + -2.25 Combine like terms: 3.010398645 + -2.25 = 0.760398645 L = 0.760398645 Simplifying L = 0.760398645

Subproblem 2

L + 2.25 = -3.010398645 Simplifying L + 2.25 = -3.010398645 Reorder the terms: 2.25 + L = -3.010398645 Solving 2.25 + L = -3.010398645 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + L = -3.010398645 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + L = -3.010398645 + -2.25 L = -3.010398645 + -2.25 Combine like terms: -3.010398645 + -2.25 = -5.260398645 L = -5.260398645 Simplifying L = -5.260398645

Solution

The solution to the problem is based on the solutions from the subproblems. L = {0.760398645, -5.260398645}

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