Calculus Examples

$3 {(t + h)}^{2} + 9 - (3 t^{2} + 9)$

Step 1

Rewrite ${(t + h)}^{2}$ as $(t + h) (t + h)$ .

$3 ((t + h) (t + h)) + 9 - (3 t^{2} + 9)$

Step 2

Expand $(t + h) (t + h)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 (t (t + h) + h (t + h)) + 9 - (3 t^{2} + 9)$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $t$ by $t$ .

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

Step 3.1.2

Multiply $h$ by $h$ .

$3 (t^{2} + t h + h t + h^{2}) + 9 - (3 t^{2} + 9)$

Step 3.2

Add $t h$ and $h t$ .

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Step 3.2.1

Reorder $t$ and $h$ .

$3 (t^{2} + h t + h t + h^{2}) + 9 - (3 t^{2} + 9)$

Step 3.2.2

Add $h t$ and $h t$ .

$3 (t^{2} + 2 h t + h^{2}) + 9 - (3 t^{2} + 9)$

Step 4

Apply the distributive property.

$3 t^{2} + 3 (2 h t) + 3 h^{2} + 9 - (3 t^{2} + 9)$

Step 5

Multiply $2$ by $3$ .

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

Step 6

Remove parentheses.

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

Step 7

Apply the distributive property.

$3 t^{2} + 6 h t + 3 h^{2} + 9 - (3 t^{2}) - 1 \cdot 9$

Step 8

Multiply $3$ by $- 1$ .

$3 t^{2} + 6 h t + 3 h^{2} + 9 - 3 t^{2} - 1 \cdot 9$

Step 9

Multiply $- 1$ by $9$ .

$3 t^{2} + 6 h t + 3 h^{2} + 9 - 3 t^{2} - 9$

Step 10

Subtract $3 t^{2}$ from $3 t^{2}$ .

$6 h t + 3 h^{2} + 9 + 0 - 9$

Step 11

Add $6 h t$ and $0$ .

$6 h t + 3 h^{2} + 9 - 9$

Step 12

Subtract $9$ from $9$ .

$6 h t + 3 h^{2} + 0$

Step 13

Add $6 h t + 3 h^{2}$ and $0$ .

$6 h t + 3 h^{2}$

Step 14

Factor $3 h$ out of $6 h t + 3 h^{2}$ .

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Step 14.1

Factor $3 h$ out of $6 h t$ .

$3 h (2 t) + 3 h^{2}$

Step 14.2

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

$3 h (2 t) + 3 h (h)$

Step 14.3

Factor $3 h$ out of $3 h (2 t) + 3 h (h)$ .

$3 h (2 t + h)$