L^2-9L+3=0

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Solution for L^2-9L+3=0 equation:


Simplifying
L2 + -9L + 3 = 0

Reorder the terms:
3 + -9L + L2 = 0

Solving
3 + -9L + L2 = 0

Solving for variable 'L'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -9L + -3 + L2 = 0 + -3

Reorder the terms:
3 + -3 + -9L + L2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -9L + L2 = 0 + -3
-9L + L2 = 0 + -3

Combine like terms: 0 + -3 = -3
-9L + L2 = -3

The L term is -9L.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9L + 20.25 + L2 = -3 + 20.25

Reorder the terms:
20.25 + -9L + L2 = -3 + 20.25

Combine like terms: -3 + 20.25 = 17.25
20.25 + -9L + L2 = 17.25

Factor a perfect square on the left side:
(L + -4.5)(L + -4.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(L + -4.5) equal to 4.153311931 and -4.153311931.

Subproblem 1

L + -4.5 = 4.153311931 Simplifying L + -4.5 = 4.153311931 Reorder the terms: -4.5 + L = 4.153311931 Solving -4.5 + L = 4.153311931 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + L = 4.153311931 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + L = 4.153311931 + 4.5 L = 4.153311931 + 4.5 Combine like terms: 4.153311931 + 4.5 = 8.653311931 L = 8.653311931 Simplifying L = 8.653311931

Subproblem 2

L + -4.5 = -4.153311931 Simplifying L + -4.5 = -4.153311931 Reorder the terms: -4.5 + L = -4.153311931 Solving -4.5 + L = -4.153311931 Solving for variable 'L'. Move all terms containing L to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + L = -4.153311931 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + L = -4.153311931 + 4.5 L = -4.153311931 + 4.5 Combine like terms: -4.153311931 + 4.5 = 0.346688069 L = 0.346688069 Simplifying L = 0.346688069

Solution

The solution to the problem is based on the solutions from the subproblems. L = {8.653311931, 0.346688069}

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