M^2-5M+11=10

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Solution for M^2-5M+11=10 equation: 


Simplifying
M2 + -5M + 11 = 10

Reorder the terms:
11 + -5M + M2 = 10

Solving
11 + -5M + M2 = 10

Solving for variable 'M'.

Reorder the terms:
11 + -10 + -5M + M2 = 10 + -10

Combine like terms: 11 + -10 = 1
1 + -5M + M2 = 10 + -10

Combine like terms: 10 + -10 = 0
1 + -5M + M2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5M + -1 + M2 = 0 + -1

Reorder the terms:
1 + -1 + -5M + M2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5M + M2 = 0 + -1
-5M + M2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5M + M2 = -1

The M term is -5M.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5M + 6.25 + M2 = -1 + 6.25

Reorder the terms:
6.25 + -5M + M2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5M + M2 = 5.25

Factor a perfect square on the left side:
(M + -2.5)(M + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(M + -2.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
M + -2.5 = 2.291287847

Simplifying
M + -2.5 = 2.291287847

Reorder the terms:
-2.5 + M = 2.291287847

Solving
-2.5 + M = 2.291287847

Solving for variable 'M'.

Move all terms containing M to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + M = 2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + M = 2.291287847 + 2.5
M = 2.291287847 + 2.5

Combine like terms: 2.291287847 + 2.5 = 4.791287847
M = 4.791287847

Simplifying
M = 4.791287847


Subproblem 2
M + -2.5 = -2.291287847

Simplifying
M + -2.5 = -2.291287847

Reorder the terms:
-2.5 + M = -2.291287847

Solving
-2.5 + M = -2.291287847

Solving for variable 'M'.

Move all terms containing M to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + M = -2.291287847 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + M = -2.291287847 + 2.5
M = -2.291287847 + 2.5

Combine like terms: -2.291287847 + 2.5 = 0.208712153
M = 0.208712153

Simplifying
M = 0.208712153


Solution
The solution to the problem is based on the solutions
from the subproblems.
M = {4.791287847, 0.208712153}

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