l^2+3l-75=0

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Solution for l^2+3l-75=0 equation:


Simplifying
l2 + 3l + -75 = 0

Reorder the terms:
-75 + 3l + l2 = 0

Solving
-75 + 3l + l2 = 0

Solving for variable 'l'.

Begin completing the square.

Move the constant term to the right:

Add '75' to each side of the equation.
-75 + 3l + 75 + l2 = 0 + 75

Reorder the terms:
-75 + 75 + 3l + l2 = 0 + 75

Combine like terms: -75 + 75 = 0
0 + 3l + l2 = 0 + 75
3l + l2 = 0 + 75

Combine like terms: 0 + 75 = 75
3l + l2 = 75

The l term is 3l.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3l + 2.25 + l2 = 75 + 2.25

Reorder the terms:
2.25 + 3l + l2 = 75 + 2.25

Combine like terms: 75 + 2.25 = 77.25
2.25 + 3l + l2 = 77.25

Factor a perfect square on the left side:
(l + 1.5)(l + 1.5) = 77.25

Calculate the square root of the right side: 8.789197916

Break this problem into two subproblems by setting 
(l + 1.5) equal to 8.789197916 and -8.789197916.

Subproblem 1

l + 1.5 = 8.789197916 Simplifying l + 1.5 = 8.789197916 Reorder the terms: 1.5 + l = 8.789197916 Solving 1.5 + l = 8.789197916 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + l = 8.789197916 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + l = 8.789197916 + -1.5 l = 8.789197916 + -1.5 Combine like terms: 8.789197916 + -1.5 = 7.289197916 l = 7.289197916 Simplifying l = 7.289197916

Subproblem 2

l + 1.5 = -8.789197916 Simplifying l + 1.5 = -8.789197916 Reorder the terms: 1.5 + l = -8.789197916 Solving 1.5 + l = -8.789197916 Solving for variable 'l'. Move all terms containing l to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + l = -8.789197916 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + l = -8.789197916 + -1.5 l = -8.789197916 + -1.5 Combine like terms: -8.789197916 + -1.5 = -10.289197916 l = -10.289197916 Simplifying l = -10.289197916

Solution

The solution to the problem is based on the solutions from the subproblems. l = {7.289197916, -10.289197916}

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