Algebra Examples

$(3 p - 2 q + 5) (3 p - 2 q - 5)$

Step 1

Expand $(3 p - 2 q + 5) (3 p - 2 q - 5)$ by multiplying each term in the first expression by each term in the second expression.

$3 p (3 p) + 3 p (- 2 q) + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 3 p \cdot p + 3 p (- 2 q) + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.2

Multiply $p$ by $p$ by adding the exponents.

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Step 2.1.2.1

Move $p$ .

$3 \cdot 3 (p \cdot p) + 3 p (- 2 q) + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.2.2

Multiply $p$ by $p$ .

$3 \cdot 3 p^{2} + 3 p (- 2 q) + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.3

Multiply $3$ by $3$ .

$9 p^{2} + 3 p (- 2 q) + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$9 p^{2} + 3 \cdot - 2 p q + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.5

Multiply $3$ by $- 2$ .

$9 p^{2} - 6 p q + 3 p \cdot - 5 - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.6

Multiply $- 5$ by $3$ .

$9 p^{2} - 6 p q - 15 p - 2 q (3 p) - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$9 p^{2} - 6 p q - 15 p - 2 \cdot 3 q p - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.8

Multiply $- 2$ by $3$ .

$9 p^{2} - 6 p q - 15 p - 6 q p - 2 q (- 2 q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.9

Rewrite using the commutative property of multiplication.

$9 p^{2} - 6 p q - 15 p - 6 q p - 2 \cdot - 2 q \cdot q - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.10

Multiply $q$ by $q$ by adding the exponents.

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Step 2.1.10.1

Move $q$ .

$9 p^{2} - 6 p q - 15 p - 6 q p - 2 \cdot - 2 (q \cdot q) - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.10.2

Multiply $q$ by $q$ .

$9 p^{2} - 6 p q - 15 p - 6 q p - 2 \cdot - 2 q^{2} - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.11

Multiply $- 2$ by $- 2$ .

$9 p^{2} - 6 p q - 15 p - 6 q p + 4 q^{2} - 2 q \cdot - 5 + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.12

Multiply $- 5$ by $- 2$ .

$9 p^{2} - 6 p q - 15 p - 6 q p + 4 q^{2} + 10 q + 5 (3 p) + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.13

Multiply $3$ by $5$ .

$9 p^{2} - 6 p q - 15 p - 6 q p + 4 q^{2} + 10 q + 15 p + 5 (- 2 q) + 5 \cdot - 5$

Step 2.1.14

Multiply $- 2$ by $5$ .

$9 p^{2} - 6 p q - 15 p - 6 q p + 4 q^{2} + 10 q + 15 p - 10 q + 5 \cdot - 5$

Step 2.1.15

Multiply $5$ by $- 5$ .

$9 p^{2} - 6 p q - 15 p - 6 q p + 4 q^{2} + 10 q + 15 p - 10 q - 25$

Step 2.2

Combine the opposite terms in $9 p^{2} - 6 p q - 15 p - 6 q p + 4 q^{2} + 10 q + 15 p - 10 q - 25$ .

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Step 2.2.1

Add $- 15 p$ and $15 p$ .

$9 p^{2} - 6 p q - 6 q p + 4 q^{2} + 10 q + 0 - 10 q - 25$

Step 2.2.2

Add $9 p^{2} - 6 p q - 6 q p + 4 q^{2} + 10 q$ and $0$ .

$9 p^{2} - 6 p q - 6 q p + 4 q^{2} + 10 q - 10 q - 25$

Step 2.2.3

Subtract $10 q$ from $10 q$ .

$9 p^{2} - 6 p q - 6 q p + 4 q^{2} + 0 - 25$

Step 2.2.4

Add $9 p^{2} - 6 p q - 6 q p + 4 q^{2}$ and $0$ .

$9 p^{2} - 6 p q - 6 q p + 4 q^{2} - 25$

Step 3

Subtract $6 q p$ from $- 6 p q$ .

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Step 3.1

Move $q$ .

$9 p^{2} - 6 p q - 6 p q + 4 q^{2} - 25$

Step 3.2

Subtract $6 p q$ from $- 6 p q$ .

$9 p^{2} - 12 p q + 4 q^{2} - 25$