L(5+2L)=46

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Solution for L(5+2L)=46 equation:


Simplifying
L(5 + 2L) = 46
(5 * L + 2L * L) = 46
(5L + 2L2) = 46

Solving
5L + 2L2 = 46

Solving for variable 'L'.

Reorder the terms:
-46 + 5L + 2L2 = 46 + -46

Combine like terms: 46 + -46 = 0
-46 + 5L + 2L2 = 0

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

Divide each side by '2'.
-23 + 2.5L + L2 = 0

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 2.5L + 23 + L2 = 0 + 23

Reorder the terms:
-23 + 23 + 2.5L + L2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 2.5L + L2 = 0 + 23
2.5L + L2 = 0 + 23

Combine like terms: 0 + 23 = 23
2.5L + L2 = 23

The L term is 2.5L.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5L + 1.5625 + L2 = 23 + 1.5625

Reorder the terms:
1.5625 + 2.5L + L2 = 23 + 1.5625

Combine like terms: 23 + 1.5625 = 24.5625
1.5625 + 2.5L + L2 = 24.5625

Factor a perfect square on the left side:
(L + 1.25)(L + 1.25) = 24.5625

Calculate the square root of the right side: 4.9560569

Break this problem into two subproblems by setting 
(L + 1.25) equal to 4.9560569 and -4.9560569.

Subproblem 1

L + 1.25 = 4.9560569 Simplifying L + 1.25 = 4.9560569 Reorder the terms: 1.25 + L = 4.9560569 Solving 1.25 + L = 4.9560569 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + L = 4.9560569 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + L = 4.9560569 + -1.25 L = 4.9560569 + -1.25 Combine like terms: 4.9560569 + -1.25 = 3.7060569 L = 3.7060569 Simplifying L = 3.7060569

Subproblem 2

L + 1.25 = -4.9560569 Simplifying L + 1.25 = -4.9560569 Reorder the terms: 1.25 + L = -4.9560569 Solving 1.25 + L = -4.9560569 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + L = -4.9560569 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + L = -4.9560569 + -1.25 L = -4.9560569 + -1.25 Combine like terms: -4.9560569 + -1.25 = -6.2060569 L = -6.2060569 Simplifying L = -6.2060569

Solution

The solution to the problem is based on the solutions from the subproblems. L = {3.7060569, -6.2060569}

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