Solve for n 0.3325=(n^2-1)/(3n^2)

$0.3325 = \frac{n^{2} - 1}{3 n^{2}}$

Rewrite the equation as $\frac{n^{2} - 1}{3 n^{2}} = 0.3325$ .

$\frac{n^{2} - 1}{3 n^{2}} = 0.3325$

Factor each term.

Rewrite $1$ as $1^{2}$ .

$\frac{n^{2} - 1^{2}}{3 n^{2}} = 0.3325$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = n$ and $b = 1$ .

$\frac{(n + 1) (n - 1)}{3 n^{2}} = 0.3325$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 n^{2}, 1$

The LCM of one and any expression is the expression.

$3 n^{2}$

Multiply each term in $\frac{(n + 1) (n - 1)}{3 n^{2}} = 0.3325$ by $3 n^{2}$ to eliminate the fractions.

Multiply each term in $\frac{(n + 1) (n - 1)}{3 n^{2}} = 0.3325$ by $3 n^{2}$ .

$\frac{(n + 1) (n - 1)}{3 n^{2}} (3 n^{2}) = 0.3325 (3 n^{2})$

Simplify the left side.

Reduce the expression by cancelling the common factors.

Rewrite using the commutative property of multiplication.

$3 \frac{(n + 1) (n - 1)}{3 n^{2}} n^{2} = 0.3325 (3 n^{2})$

Cancel the common factor of $3$ .

Factor $3$ out of $3 n^{2}$ .

$3 \frac{(n + 1) (n - 1)}{3 (n^{2})} n^{2} = 0.3325 (3 n^{2})$

Cancel the common factor.

$3 \frac{(n + 1) (n - 1)}{3 n^{2}} n^{2} = 0.3325 (3 n^{2})$

Rewrite the expression.

$\frac{(n + 1) (n - 1)}{n^{2}} n^{2} = 0.3325 (3 n^{2})$

Cancel the common factor of $n^{2}$ .

Cancel the common factor.

$\frac{(n + 1) (n - 1)}{n^{2}} n^{2} = 0.3325 (3 n^{2})$

Rewrite the expression.

$(n + 1) (n - 1) = 0.3325 (3 n^{2})$

Expand $(n + 1) (n - 1)$ using the FOIL Method.

Apply the distributive property.

$n (n - 1) + 1 (n - 1) = 0.3325 (3 n^{2})$

Apply the distributive property.

$n \cdot n + n \cdot - 1 + 1 (n - 1) = 0.3325 (3 n^{2})$

Apply the distributive property.

$n \cdot n + n \cdot - 1 + 1 n + 1 \cdot - 1 = 0.3325 (3 n^{2})$

Simplify and combine like terms.

Simplify each term.

Multiply $n$ by $n$ .

$n^{2} + n \cdot - 1 + 1 n + 1 \cdot - 1 = 0.3325 (3 n^{2})$

Move $- 1$ to the left of $n$ .

$n^{2} - 1 \cdot n + 1 n + 1 \cdot - 1 = 0.3325 (3 n^{2})$

Rewrite $- 1 n$ as $- n$ .

$n^{2} - n + 1 n + 1 \cdot - 1 = 0.3325 (3 n^{2})$

Multiply $n$ by $1$ .

$n^{2} - n + n + 1 \cdot - 1 = 0.3325 (3 n^{2})$

Multiply $- 1$ by $1$ .

$n^{2} - n + n - 1 = 0.3325 (3 n^{2})$

Add $- n$ and $n$ .

$n^{2} + 0 - 1 = 0.3325 (3 n^{2})$

Add $n^{2}$ and $0$ .

$n^{2} - 1 = 0.3325 (3 n^{2})$

Simplify the right side.

Multiply $3$ by $0.3325$ .

$n^{2} - 1 = 0.9975 n^{2}$

Solve the equation.

Move all terms containing $n$ to the left side of the equation.

Subtract $0.9975 n^{2}$ from both sides of the equation.

$n^{2} - 1 - 0.9975 n^{2} = 0$

Subtract $0.9975 n^{2}$ from $n^{2}$ .

$0.0025 n^{2} - 1 = 0$

Add $1$ to both sides of the equation.

$0.0025 n^{2} = 1$

Divide each term in $0.0025 n^{2} = 1$ by $0.0025$ and simplify.

Divide each term in $0.0025 n^{2} = 1$ by $0.0025$ .

$\frac{0.0025 n^{2}}{0.0025} = \frac{1}{0.0025}$

Simplify the left side.

Cancel the common factor of $0.0025$ .

Cancel the common factor.

$\frac{0.0025 n^{2}}{0.0025} = \frac{1}{0.0025}$

Divide $n^{2}$ by $1$ .

$n^{2} = \frac{1}{0.0025}$

Simplify the right side.

Divide $1$ by $0.0025$ .

$n^{2} = 400$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$n = \pm \sqrt{400}$

Simplify $\pm \sqrt{400}$ .

Rewrite $400$ as $20^{2}$ .

$n = \pm \sqrt{20^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$n = \pm 20$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$n = 20$

Next, use the negative value of the $\pm$ to find the second solution.

$n = - 20$

The complete solution is the result of both the positive and negative portions of the solution.

$n = 20, - 20$

Solve for n 0.3325=(n^2-1)/(3n^2)

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