Algebra Examples

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

Step 1

Factor each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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 $b = 1$ .

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

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

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

$3 n^{2}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$3 n^{2}$

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

Cancel the common factor.

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

Step 3.2.1.2.3

Rewrite the expression.

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

Step 3.2.1.3

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

Tap for more steps...

Step 3.2.1.3.1

Cancel the common factor.

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

Step 3.2.1.3.2

Rewrite the expression.

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

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

Apply the distributive property.

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

Step 3.2.2.2

Apply the distributive property.

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

Step 3.2.2.3

Apply the distributive property.

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

Step 3.2.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Multiply $n$ by $n$ .

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

Step 3.2.3.1.2

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

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

Step 3.2.3.1.3

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

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

Step 3.2.3.1.4

Multiply $n$ by $1$ .

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

Step 3.2.3.1.5

Multiply $- 1$ by $1$ .

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

Step 3.2.3.2

Add $- n$ and $n$ .

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

Step 3.2.3.3

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

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

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $3$ by $0.3325$ .

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

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Add $1$ to both sides of the equation.

$0.0025 n^{2} = 1$

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $0.0025$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

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

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

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Divide $1$ by $0.0025$ .

$n^{2} = 400$

Step 4.4

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

$n = \pm \sqrt{400}$

Step 4.5

Simplify $\pm \sqrt{400}$ .

Tap for more steps...

Step 4.5.1

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

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

Step 4.5.2

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

$n = \pm 20$

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

$n = 20$

Step 4.6.2

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

$n = - 20$

Step 4.6.3

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

$n = 20, - 20$