9n^2=4n+9

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Solution for 9n^2=4n+9 equation:


Simplifying
9n2 = 4n + 9

Reorder the terms:
9n2 = 9 + 4n

Solving
9n2 = 9 + 4n

Solving for variable 'n'.

Reorder the terms:
-9 + -4n + 9n2 = 9 + 4n + -9 + -4n

Reorder the terms:
-9 + -4n + 9n2 = 9 + -9 + 4n + -4n

Combine like terms: 9 + -9 = 0
-9 + -4n + 9n2 = 0 + 4n + -4n
-9 + -4n + 9n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
-9 + -4n + 9n2 = 0

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

Divide each side by '9'.
-1 + -0.4444444444n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.4444444444n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.4444444444n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.4444444444n + n2 = 0 + 1
-0.4444444444n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.4444444444n + n2 = 1

The n term is -0.4444444444n.  Take half its coefficient (-0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
-0.4444444444n + 0.04938271604 + n2 = 1 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444n + n2 = 1 + 0.04938271604

Combine like terms: 1 + 0.04938271604 = 1.04938271604
0.04938271604 + -0.4444444444n + n2 = 1.04938271604

Factor a perfect square on the left side:
(n + -0.2222222222)(n + -0.2222222222) = 1.04938271604

Calculate the square root of the right side: 1.024393829

Break this problem into two subproblems by setting 
(n + -0.2222222222) equal to 1.024393829 and -1.024393829.

Subproblem 1

n + -0.2222222222 = 1.024393829 Simplifying n + -0.2222222222 = 1.024393829 Reorder the terms: -0.2222222222 + n = 1.024393829 Solving -0.2222222222 + n = 1.024393829 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + n = 1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + n = 1.024393829 + 0.2222222222 n = 1.024393829 + 0.2222222222 Combine like terms: 1.024393829 + 0.2222222222 = 1.2466160512 n = 1.2466160512 Simplifying n = 1.2466160512

Subproblem 2

n + -0.2222222222 = -1.024393829 Simplifying n + -0.2222222222 = -1.024393829 Reorder the terms: -0.2222222222 + n = -1.024393829 Solving -0.2222222222 + n = -1.024393829 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + n = -1.024393829 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + n = -1.024393829 + 0.2222222222 n = -1.024393829 + 0.2222222222 Combine like terms: -1.024393829 + 0.2222222222 = -0.8021716068 n = -0.8021716068 Simplifying n = -0.8021716068

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.2466160512, -0.8021716068}

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