78=2m(m+3)+m

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Solution for 78=2m(m+3)+m equation:


Simplifying
78 = 2m(m + 3) + m

Reorder the terms:
78 = 2m(3 + m) + m
78 = (3 * 2m + m * 2m) + m
78 = (6m + 2m2) + m

Reorder the terms:
78 = 6m + m + 2m2

Combine like terms: 6m + m = 7m
78 = 7m + 2m2

Solving
78 = 7m + 2m2

Solving for variable 'm'.

Reorder the terms:
78 + -7m + -2m2 = 7m + -7m + 2m2 + -2m2

Combine like terms: 7m + -7m = 0
78 + -7m + -2m2 = 0 + 2m2 + -2m2
78 + -7m + -2m2 = 2m2 + -2m2

Combine like terms: 2m2 + -2m2 = 0
78 + -7m + -2m2 = 0

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

Divide each side by '-2'.
-39 + 3.5m + m2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 3.5m + 39 + m2 = 0 + 39

Reorder the terms:
-39 + 39 + 3.5m + m2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 3.5m + m2 = 0 + 39
3.5m + m2 = 0 + 39

Combine like terms: 0 + 39 = 39
3.5m + m2 = 39

The m term is 3.5m.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5m + 3.0625 + m2 = 39 + 3.0625

Reorder the terms:
3.0625 + 3.5m + m2 = 39 + 3.0625

Combine like terms: 39 + 3.0625 = 42.0625
3.0625 + 3.5m + m2 = 42.0625

Factor a perfect square on the left side:
(m + 1.75)(m + 1.75) = 42.0625

Calculate the square root of the right side: 6.485560886

Break this problem into two subproblems by setting 
(m + 1.75) equal to 6.485560886 and -6.485560886.

Subproblem 1

m + 1.75 = 6.485560886 Simplifying m + 1.75 = 6.485560886 Reorder the terms: 1.75 + m = 6.485560886 Solving 1.75 + m = 6.485560886 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + m = 6.485560886 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + m = 6.485560886 + -1.75 m = 6.485560886 + -1.75 Combine like terms: 6.485560886 + -1.75 = 4.735560886 m = 4.735560886 Simplifying m = 4.735560886

Subproblem 2

m + 1.75 = -6.485560886 Simplifying m + 1.75 = -6.485560886 Reorder the terms: 1.75 + m = -6.485560886 Solving 1.75 + m = -6.485560886 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + m = -6.485560886 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + m = -6.485560886 + -1.75 m = -6.485560886 + -1.75 Combine like terms: -6.485560886 + -1.75 = -8.235560886 m = -8.235560886 Simplifying m = -8.235560886

Solution

The solution to the problem is based on the solutions from the subproblems. m = {4.735560886, -8.235560886}

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