Algebra Examples

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

Step 1

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

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

Step 2

Factor each term.

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Step 2.1

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

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

Step 2.2

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 3

Find the LCD of the terms in the equation.

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Step 3.1

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 3.2

The LCM of one and any expression is the expression.

$3 n^{2}$

Step 4

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

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Step 4.1

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 4.2

Simplify the left side.

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Step 4.2.1

Reduce the expression by cancelling the common factors.

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Step 4.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Cancel the common factor.

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

Step 4.2.1.2.3

Rewrite the expression.

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

Cancel the common factor.

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

Step 4.2.1.3.2

Rewrite the expression.

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

Step 4.2.2

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

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Step 4.2.2.1

Apply the distributive property.

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

Step 4.2.2.2

Apply the distributive property.

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

Step 4.2.2.3

Apply the distributive property.

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

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply $n$ by $n$ .

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

Step 4.2.3.1.2

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

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

Step 4.2.3.1.3

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

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

Step 4.2.3.1.4

Multiply $n$ by $1$ .

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

Step 4.2.3.1.5

Multiply $- 1$ by $1$ .

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

Step 4.2.3.2

Add $- n$ and $n$ .

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

Step 4.2.3.3

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

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $3$ by $0.3325$ .

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

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.2

Add $1$ to both sides of the equation.

$0.0025 n^{2} = 1$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $0.0025$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

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

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $1$ by $0.0025$ .

$n^{2} = 400$

Step 5.4

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

$n = \pm \sqrt{400}$

Step 5.5

Simplify $\pm \sqrt{400}$ .

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Step 5.5.1

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

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

Step 5.5.2

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

$n = \pm 20$

Step 5.6

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

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Step 5.6.1

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

$n = 20$

Step 5.6.2

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

$n = - 20$

Step 5.6.3

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

$n = 20, - 20$