Algebra Examples

$\frac{n^{2} + 24 n + 144}{n^{2} + 11 n - 12} \cdot \frac{n^{2} + 2 n - 3}{A} = 1$

Step 1

Simplify $\frac{n^{2} + 24 n + 144}{n^{2} + 11 n - 12} \cdot \frac{n^{2} + 2 n - 3}{A}$ .

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $144$ as $12^{2}$ .

$\frac{n^{2} + 24 n + 12^{2}}{n^{2} + 11 n - 12} \cdot \frac{n^{2} + 2 n - 3}{A} = 1$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$24 n = 2 \cdot n \cdot 12$

Step 1.1.3

Rewrite the polynomial.

$\frac{n^{2} + 2 \cdot n \cdot 12 + 12^{2}}{n^{2} + 11 n - 12} \cdot \frac{n^{2} + 2 n - 3}{A} = 1$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = n$ and $b = 12$ .

$\frac{{(n + 12)}^{2}}{n^{2} + 11 n - 12} \cdot \frac{n^{2} + 2 n - 3}{A} = 1$

Step 1.2

Factor $n^{2} + 11 n - 12$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $11$ .

$- 1, 12$

Step 1.2.2

Write the factored form using these integers.

$\frac{{(n + 12)}^{2}}{(n - 1) (n + 12)} \cdot \frac{n^{2} + 2 n - 3}{A} = 1$

Step 1.3

Factor $n^{2} + 2 n - 3$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.3.2

Write the factored form using these integers.

$\frac{{(n + 12)}^{2}}{(n - 1) (n + 12)} \cdot \frac{(n - 1) (n + 3)}{A} = 1$

Step 1.4

Simplify terms.

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Step 1.4.1

Combine.

$\frac{{(n + 12)}^{2} ((n - 1) (n + 3))}{(n - 1) (n + 12) A} = 1$

Step 1.4.2

Cancel the common factor of ${(n + 12)}^{2}$ and $n + 12$ .

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Step 1.4.2.1

Factor $n + 12$ out of ${(n + 12)}^{2} ((n - 1) (n + 3))$ .

$\frac{(n + 12) ((n + 12) ((n - 1) (n + 3)))}{(n - 1) (n + 12) A} = 1$

Step 1.4.2.2

Cancel the common factors.

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Step 1.4.2.2.1

Factor $n + 12$ out of $(n - 1) (n + 12) A$ .

$\frac{(n + 12) ((n + 12) ((n - 1) (n + 3)))}{(n + 12) ((n - 1) A)} = 1$

Step 1.4.2.2.2

Cancel the common factor.

$\frac{(n + 12) ((n + 12) ((n - 1) (n + 3)))}{(n + 12) ((n - 1) A)} = 1$

Step 1.4.2.2.3

Rewrite the expression.

$\frac{(n + 12) ((n - 1) (n + 3))}{(n - 1) A} = 1$

Step 1.4.3

Cancel the common factor of $n - 1$ .

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Step 1.4.3.1

Cancel the common factor.

$\frac{(n + 12) ((n - 1) (n + 3))}{(n - 1) A} = 1$

Step 1.4.3.2

Rewrite the expression.

$\frac{(n + 12) (n + 3)}{A} = 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$A, 1$

Step 2.2

The LCM of one and any expression is the expression.

$A$

Step 3

Multiply each term in $\frac{(n + 12) (n + 3)}{A} = 1$ by $A$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{(n + 12) (n + 3)}{A} = 1$ by $A$ .

$\frac{(n + 12) (n + 3)}{A} A = 1 A$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $A$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(n + 12) (n + 3)}{A} A = 1 A$

Step 3.2.1.2

Rewrite the expression.

$(n + 12) (n + 3) = 1 A$

Step 3.2.2

Expand $(n + 12) (n + 3)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$n (n + 3) + 12 (n + 3) = 1 A$

Step 3.2.2.2

Apply the distributive property.

$n \cdot n + n \cdot 3 + 12 (n + 3) = 1 A$

Step 3.2.2.3

Apply the distributive property.

$n \cdot n + n \cdot 3 + 12 n + 12 \cdot 3 = 1 A$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $n$ by $n$ .

$n^{2} + n \cdot 3 + 12 n + 12 \cdot 3 = 1 A$

Step 3.2.3.1.2

Move $3$ to the left of $n$ .

$n^{2} + 3 \cdot n + 12 n + 12 \cdot 3 = 1 A$

Step 3.2.3.1.3

Multiply $12$ by $3$ .

$n^{2} + 3 n + 12 n + 36 = 1 A$

Step 3.2.3.2

Add $3 n$ and $12 n$ .

$n^{2} + 15 n + 36 = 1 A$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $A$ by $1$ .

$n^{2} + 15 n + 36 = A$

Step 4

Rewrite the equation as $A = n^{2} + 15 n + 36$ .

$A = n^{2} + 15 n + 36$