Algebra Examples

$\frac{t^{2} - 36}{3} + \frac{t^{2} + 6 t}{9 t}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{t^{2} - 6^{2}}{3} + \frac{t^{2} + 6 t}{9 t}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 6$ .

$\frac{(t + 6) (t - 6)}{3} + \frac{t^{2} + 6 t}{9 t}$

Step 1.2

Factor $t$ out of $t^{2} + 6 t$ .

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Step 1.2.1

Factor $t$ out of $t^{2}$ .

$\frac{(t + 6) (t - 6)}{3} + \frac{t \cdot t + 6 t}{9 t}$

Step 1.2.2

Factor $t$ out of $6 t$ .

$\frac{(t + 6) (t - 6)}{3} + \frac{t \cdot t + t \cdot 6}{9 t}$

Step 1.2.3

Factor $t$ out of $t \cdot t + t \cdot 6$ .

$\frac{(t + 6) (t - 6)}{3} + \frac{t (t + 6)}{9 t}$

Step 1.3

Cancel the common factor of $t$ .

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Step 1.3.1

Cancel the common factor.

$\frac{(t + 6) (t - 6)}{3} + \frac{t (t + 6)}{9 t}$

Step 1.3.2

Rewrite the expression.

$\frac{(t + 6) (t - 6)}{3} + \frac{t + 6}{9}$

Step 2

To write $\frac{(t + 6) (t - 6)}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{(t + 6) (t - 6)}{3} \cdot \frac{3}{3} + \frac{t + 6}{9}$

Step 3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{(t + 6) (t - 6)}{3}$ by $\frac{3}{3}$ .

$\frac{(t + 6) (t - 6) \cdot 3}{3 \cdot 3} + \frac{t + 6}{9}$

Step 3.2

Multiply $3$ by $3$ .

$\frac{(t + 6) (t - 6) \cdot 3}{9} + \frac{t + 6}{9}$

Step 4

Combine the numerators over the common denominator.

$\frac{(t + 6) (t - 6) \cdot 3 + t + 6}{9}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(t + 6) (t - 6)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{(t (t - 6) + 6 (t - 6)) \cdot 3 + t + 6}{9}$

Step 5.1.2

Apply the distributive property.

$\frac{(t \cdot t + t \cdot - 6 + 6 (t - 6)) \cdot 3 + t + 6}{9}$

Step 5.1.3

Apply the distributive property.

$\frac{(t \cdot t + t \cdot - 6 + 6 t + 6 \cdot - 6) \cdot 3 + t + 6}{9}$

Step 5.2

Combine the opposite terms in $t \cdot t + t \cdot - 6 + 6 t + 6 \cdot - 6$ .

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Step 5.2.1

Reorder the factors in the terms $t \cdot - 6$ and $6 t$ .

$\frac{(t \cdot t - 6 t + 6 t + 6 \cdot - 6) \cdot 3 + t + 6}{9}$

Step 5.2.2

Add $- 6 t$ and $6 t$ .

$\frac{(t \cdot t + 0 + 6 \cdot - 6) \cdot 3 + t + 6}{9}$

Step 5.2.3

Add $t \cdot t$ and $0$ .

$\frac{(t \cdot t + 6 \cdot - 6) \cdot 3 + t + 6}{9}$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $t$ by $t$ .

$\frac{(t^{2} + 6 \cdot - 6) \cdot 3 + t + 6}{9}$

Step 5.3.2

Multiply $6$ by $- 6$ .

$\frac{(t^{2} - 36) \cdot 3 + t + 6}{9}$

Step 5.4

Apply the distributive property.

$\frac{t^{2} \cdot 3 - 36 \cdot 3 + t + 6}{9}$

Step 5.5

Move $3$ to the left of $t^{2}$ .

$\frac{3 \cdot t^{2} - 36 \cdot 3 + t + 6}{9}$

Step 5.6

Multiply $- 36$ by $3$ .

$\frac{3 \cdot t^{2} - 108 + t + 6}{9}$

Step 5.7

Add $- 108$ and $6$ .

$\frac{3 t^{2} + t - 102}{9}$

Step 5.8

Factor by grouping.

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Step 5.8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 102 = - 306$ and whose sum is $b = 1$ .

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Step 5.8.1.1

Multiply by $1$ .

$\frac{3 t^{2} + 1 t - 102}{9}$

Step 5.8.1.2

Rewrite $1$ as $- 17$ plus $18$

$\frac{3 t^{2} + (- 17 + 18) t - 102}{9}$

Step 5.8.1.3

Apply the distributive property.

$\frac{3 t^{2} - 17 t + 18 t - 102}{9}$

Step 5.8.2

Factor out the greatest common factor from each group.

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Step 5.8.2.1

Group the first two terms and the last two terms.

$\frac{(3 t^{2} - 17 t) + 18 t - 102}{9}$

Step 5.8.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{t (3 t - 17) + 6 (3 t - 17)}{9}$

Step 5.8.3

Factor the polynomial by factoring out the greatest common factor, $3 t - 17$ .

$\frac{(3 t - 17) (t + 6)}{9}$