l(3+.5l)=140

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Solution for l(3+.5l)=140 equation:


Simplifying
l(3 + 0.5l) = 140
(3 * l + 0.5l * l) = 140
(3l + 0.5l2) = 140

Solving
3l + 0.5l2 = 140

Solving for variable 'l'.

Reorder the terms:
-140 + 3l + 0.5l2 = 140 + -140

Combine like terms: 140 + -140 = 0
-140 + 3l + 0.5l2 = 0

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

Divide each side by '0.5'.
-280 + 6l + l2 = 0

Move the constant term to the right:

Add '280' to each side of the equation.
-280 + 6l + 280 + l2 = 0 + 280

Reorder the terms:
-280 + 280 + 6l + l2 = 0 + 280

Combine like terms: -280 + 280 = 0
0 + 6l + l2 = 0 + 280
6l + l2 = 0 + 280

Combine like terms: 0 + 280 = 280
6l + l2 = 280

The l term is 6l.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6l + 9 + l2 = 280 + 9

Reorder the terms:
9 + 6l + l2 = 280 + 9

Combine like terms: 280 + 9 = 289
9 + 6l + l2 = 289

Factor a perfect square on the left side:
(l + 3)(l + 3) = 289

Calculate the square root of the right side: 17

Break this problem into two subproblems by setting 
(l + 3) equal to 17 and -17.

Subproblem 1

l + 3 = 17 Simplifying l + 3 = 17 Reorder the terms: 3 + l = 17 Solving 3 + l = 17 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + l = 17 + -3 Combine like terms: 3 + -3 = 0 0 + l = 17 + -3 l = 17 + -3 Combine like terms: 17 + -3 = 14 l = 14 Simplifying l = 14

Subproblem 2

l + 3 = -17 Simplifying l + 3 = -17 Reorder the terms: 3 + l = -17 Solving 3 + l = -17 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + l = -17 + -3 Combine like terms: 3 + -3 = 0 0 + l = -17 + -3 l = -17 + -3 Combine like terms: -17 + -3 = -20 l = -20 Simplifying l = -20

Solution

The solution to the problem is based on the solutions from the subproblems. l = {14, -20}

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