m(2m+3)=1

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Solution for m(2m+3)=1 equation:


Simplifying
m(2m + 3) = 1

Reorder the terms:
m(3 + 2m) = 1
(3 * m + 2m * m) = 1
(3m + 2m2) = 1

Solving
3m + 2m2 = 1

Solving for variable 'm'.

Reorder the terms:
-1 + 3m + 2m2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 3m + 2m2 = 0

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

Divide each side by '2'.
-0.5 + 1.5m + m2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 1.5m + 0.5 + m2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 1.5m + m2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 1.5m + m2 = 0 + 0.5
1.5m + m2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
1.5m + m2 = 0.5

The m term is 1.5m.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5m + 0.5625 + m2 = 0.5 + 0.5625

Reorder the terms:
0.5625 + 1.5m + m2 = 0.5 + 0.5625

Combine like terms: 0.5 + 0.5625 = 1.0625
0.5625 + 1.5m + m2 = 1.0625

Factor a perfect square on the left side:
(m + 0.75)(m + 0.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(m + 0.75) equal to 1.030776406 and -1.030776406.

Subproblem 1

m + 0.75 = 1.030776406 Simplifying m + 0.75 = 1.030776406 Reorder the terms: 0.75 + m = 1.030776406 Solving 0.75 + m = 1.030776406 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + m = 1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + m = 1.030776406 + -0.75 m = 1.030776406 + -0.75 Combine like terms: 1.030776406 + -0.75 = 0.280776406 m = 0.280776406 Simplifying m = 0.280776406

Subproblem 2

m + 0.75 = -1.030776406 Simplifying m + 0.75 = -1.030776406 Reorder the terms: 0.75 + m = -1.030776406 Solving 0.75 + m = -1.030776406 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + m = -1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + m = -1.030776406 + -0.75 m = -1.030776406 + -0.75 Combine like terms: -1.030776406 + -0.75 = -1.780776406 m = -1.780776406 Simplifying m = -1.780776406

Solution

The solution to the problem is based on the solutions from the subproblems. m = {0.280776406, -1.780776406}

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