3m^2+5(1+4m)=m+22+m^2

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Solution for 3m^2+5(1+4m)=m+22+m^2 equation:


Simplifying
3m2 + 5(1 + 4m) = m + 22 + m2
3m2 + (1 * 5 + 4m * 5) = m + 22 + m2
3m2 + (5 + 20m) = m + 22 + m2

Reorder the terms:
5 + 20m + 3m2 = m + 22 + m2

Reorder the terms:
5 + 20m + 3m2 = 22 + m + m2

Solving
5 + 20m + 3m2 = 22 + m + m2

Solving for variable 'm'.

Reorder the terms:
5 + -22 + 20m + -1m + 3m2 + -1m2 = 22 + m + m2 + -22 + -1m + -1m2

Combine like terms: 5 + -22 = -17
-17 + 20m + -1m + 3m2 + -1m2 = 22 + m + m2 + -22 + -1m + -1m2

Combine like terms: 20m + -1m = 19m
-17 + 19m + 3m2 + -1m2 = 22 + m + m2 + -22 + -1m + -1m2

Combine like terms: 3m2 + -1m2 = 2m2
-17 + 19m + 2m2 = 22 + m + m2 + -22 + -1m + -1m2

Reorder the terms:
-17 + 19m + 2m2 = 22 + -22 + m + -1m + m2 + -1m2

Combine like terms: 22 + -22 = 0
-17 + 19m + 2m2 = 0 + m + -1m + m2 + -1m2
-17 + 19m + 2m2 = m + -1m + m2 + -1m2

Combine like terms: m + -1m = 0
-17 + 19m + 2m2 = 0 + m2 + -1m2
-17 + 19m + 2m2 = m2 + -1m2

Combine like terms: m2 + -1m2 = 0
-17 + 19m + 2m2 = 0

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

Divide each side by '2'.
-8.5 + 9.5m + m2 = 0

Move the constant term to the right:

Add '8.5' to each side of the equation.
-8.5 + 9.5m + 8.5 + m2 = 0 + 8.5

Reorder the terms:
-8.5 + 8.5 + 9.5m + m2 = 0 + 8.5

Combine like terms: -8.5 + 8.5 = 0.0
0.0 + 9.5m + m2 = 0 + 8.5
9.5m + m2 = 0 + 8.5

Combine like terms: 0 + 8.5 = 8.5
9.5m + m2 = 8.5

The m term is 9.5m.  Take half its coefficient (4.75).
Square it (22.5625) and add it to both sides.

Add '22.5625' to each side of the equation.
9.5m + 22.5625 + m2 = 8.5 + 22.5625

Reorder the terms:
22.5625 + 9.5m + m2 = 8.5 + 22.5625

Combine like terms: 8.5 + 22.5625 = 31.0625
22.5625 + 9.5m + m2 = 31.0625

Factor a perfect square on the left side:
(m + 4.75)(m + 4.75) = 31.0625

Calculate the square root of the right side: 5.573374202

Break this problem into two subproblems by setting 
(m + 4.75) equal to 5.573374202 and -5.573374202.

Subproblem 1

m + 4.75 = 5.573374202 Simplifying m + 4.75 = 5.573374202 Reorder the terms: 4.75 + m = 5.573374202 Solving 4.75 + m = 5.573374202 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + m = 5.573374202 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + m = 5.573374202 + -4.75 m = 5.573374202 + -4.75 Combine like terms: 5.573374202 + -4.75 = 0.823374202 m = 0.823374202 Simplifying m = 0.823374202

Subproblem 2

m + 4.75 = -5.573374202 Simplifying m + 4.75 = -5.573374202 Reorder the terms: 4.75 + m = -5.573374202 Solving 4.75 + m = -5.573374202 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-4.75' to each side of the equation. 4.75 + -4.75 + m = -5.573374202 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + m = -5.573374202 + -4.75 m = -5.573374202 + -4.75 Combine like terms: -5.573374202 + -4.75 = -10.323374202 m = -10.323374202 Simplifying m = -10.323374202

Solution

The solution to the problem is based on the solutions from the subproblems. m = {0.823374202, -10.323374202}

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