Basic Math Examples

$\frac{{(z - 2)}^{2}}{4} + \frac{{(q - 3)}^{2}}{6}$

Step 1

To write $\frac{{(z - 2)}^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{{(z - 2)}^{2}}{4} \cdot \frac{3}{3} + \frac{{(q - 3)}^{2}}{6}$

Step 2

To write $\frac{{(q - 3)}^{2}}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{{(z - 2)}^{2}}{4} \cdot \frac{3}{3} + \frac{{(q - 3)}^{2}}{6} \cdot \frac{2}{2}$

Step 3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{{(z - 2)}^{2}}{4}$ by $\frac{3}{3}$ .

$\frac{{(z - 2)}^{2} \cdot 3}{4 \cdot 3} + \frac{{(q - 3)}^{2}}{6} \cdot \frac{2}{2}$

Step 3.2

Multiply $4$ by $3$ .

$\frac{{(z - 2)}^{2} \cdot 3}{12} + \frac{{(q - 3)}^{2}}{6} \cdot \frac{2}{2}$

Step 3.3

Multiply $\frac{{(q - 3)}^{2}}{6}$ by $\frac{2}{2}$ .

$\frac{{(z - 2)}^{2} \cdot 3}{12} + \frac{{(q - 3)}^{2} \cdot 2}{6 \cdot 2}$

Step 3.4

Multiply $6$ by $2$ .

$\frac{{(z - 2)}^{2} \cdot 3}{12} + \frac{{(q - 3)}^{2} \cdot 2}{12}$

Step 4

Combine the numerators over the common denominator.

$\frac{{(z - 2)}^{2} \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5

Rewrite $\frac{{(z - 2)}^{2} \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$ in a factored form.

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Step 5.1

Rewrite ${(z - 2)}^{2}$ as $(z - 2) (z - 2)$ .

$\frac{(z - 2) (z - 2) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.2

Expand $(z - 2) (z - 2)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$\frac{(z (z - 2) - 2 (z - 2)) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.2.2

Apply the distributive property.

$\frac{(z \cdot z + z \cdot - 2 - 2 (z - 2)) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.2.3

Apply the distributive property.

$\frac{(z \cdot z + z \cdot - 2 - 2 z - 2 \cdot - 2) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $z$ by $z$ .

$\frac{(z^{2} + z \cdot - 2 - 2 z - 2 \cdot - 2) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.3.1.2

Move $- 2$ to the left of $z$ .

$\frac{(z^{2} - 2 z - 2 z - 2 \cdot - 2) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{(z^{2} - 2 z - 2 z + 4) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.3.2

Subtract $2 z$ from $- 2 z$ .

$\frac{(z^{2} - 4 z + 4) \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.4

Apply the distributive property.

$\frac{z^{2} \cdot 3 - 4 z \cdot 3 + 4 \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.5

Simplify.

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Step 5.5.1

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

$\frac{3 \cdot z^{2} - 4 z \cdot 3 + 4 \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.5.2

Multiply $3$ by $- 4$ .

$\frac{3 \cdot z^{2} - 12 z + 4 \cdot 3 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.5.3

Multiply $4$ by $3$ .

$\frac{3 \cdot z^{2} - 12 z + 12 + {(q - 3)}^{2} \cdot 2}{12}$

$\frac{3 z^{2} - 12 z + 12 + {(q - 3)}^{2} \cdot 2}{12}$

Step 5.6

Rewrite ${(q - 3)}^{2}$ as $(q - 3) (q - 3)$ .

$\frac{3 z^{2} - 12 z + 12 + (q - 3) (q - 3) \cdot 2}{12}$

Step 5.7

Expand $(q - 3) (q - 3)$ using the FOIL Method.

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Step 5.7.1

Apply the distributive property.

$\frac{3 z^{2} - 12 z + 12 + (q (q - 3) - 3 (q - 3)) \cdot 2}{12}$

Step 5.7.2

Apply the distributive property.

$\frac{3 z^{2} - 12 z + 12 + (q \cdot q + q \cdot - 3 - 3 (q - 3)) \cdot 2}{12}$

Step 5.7.3

Apply the distributive property.

$\frac{3 z^{2} - 12 z + 12 + (q \cdot q + q \cdot - 3 - 3 q - 3 \cdot - 3) \cdot 2}{12}$

Step 5.8

Simplify and combine like terms.

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Step 5.8.1

Simplify each term.

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Step 5.8.1.1

Multiply $q$ by $q$ .

$\frac{3 z^{2} - 12 z + 12 + (q^{2} + q \cdot - 3 - 3 q - 3 \cdot - 3) \cdot 2}{12}$

Step 5.8.1.2

Move $- 3$ to the left of $q$ .

$\frac{3 z^{2} - 12 z + 12 + (q^{2} - 3 q - 3 q - 3 \cdot - 3) \cdot 2}{12}$

Step 5.8.1.3

Multiply $- 3$ by $- 3$ .

$\frac{3 z^{2} - 12 z + 12 + (q^{2} - 3 q - 3 q + 9) \cdot 2}{12}$

Step 5.8.2

Subtract $3 q$ from $- 3 q$ .

$\frac{3 z^{2} - 12 z + 12 + (q^{2} - 6 q + 9) \cdot 2}{12}$

Step 5.9

Apply the distributive property.

$\frac{3 z^{2} - 12 z + 12 + q^{2} \cdot 2 - 6 q \cdot 2 + 9 \cdot 2}{12}$

Step 5.10

Simplify.

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Step 5.10.1

Move $2$ to the left of $q^{2}$ .

$\frac{3 z^{2} - 12 z + 12 + 2 \cdot q^{2} - 6 q \cdot 2 + 9 \cdot 2}{12}$

Step 5.10.2

Multiply $2$ by $- 6$ .

$\frac{3 z^{2} - 12 z + 12 + 2 \cdot q^{2} - 12 q + 9 \cdot 2}{12}$

Step 5.10.3

Multiply $9$ by $2$ .

$\frac{3 z^{2} - 12 z + 12 + 2 \cdot q^{2} - 12 q + 18}{12}$

$\frac{3 z^{2} - 12 z + 12 + 2 q^{2} - 12 q + 18}{12}$

Step 5.11

Add $12$ and $18$ .

$\frac{3 z^{2} - 12 z + 2 q^{2} - 12 q + 30}{12}$