2n^2-3n=10

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Solution for 2n^2-3n=10 equation:


Simplifying
2n2 + -3n = 10

Reorder the terms:
-3n + 2n2 = 10

Solving
-3n + 2n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + -3n + 2n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -3n + 2n2 = 0

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

Divide each side by '2'.
-5 + -1.5n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1.5n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -1.5n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1.5n + n2 = 0 + 5
-1.5n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1.5n + n2 = 5

The n term is -1.5n.  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.
-1.5n + 0.5625 + n2 = 5 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 5 + 0.5625

Combine like terms: 5 + 0.5625 = 5.5625
0.5625 + -1.5n + n2 = 5.5625

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(n + -0.75) equal to 2.358495283 and -2.358495283.

Subproblem 1

n + -0.75 = 2.358495283 Simplifying n + -0.75 = 2.358495283 Reorder the terms: -0.75 + n = 2.358495283 Solving -0.75 + n = 2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 2.358495283 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 2.358495283 + 0.75 n = 2.358495283 + 0.75 Combine like terms: 2.358495283 + 0.75 = 3.108495283 n = 3.108495283 Simplifying n = 3.108495283

Subproblem 2

n + -0.75 = -2.358495283 Simplifying n + -0.75 = -2.358495283 Reorder the terms: -0.75 + n = -2.358495283 Solving -0.75 + n = -2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -2.358495283 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -2.358495283 + 0.75 n = -2.358495283 + 0.75 Combine like terms: -2.358495283 + 0.75 = -1.608495283 n = -1.608495283 Simplifying n = -1.608495283

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.108495283, -1.608495283}

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