Basic Math Examples

${(2 z - y + 2)}^{2}$

Step 1

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

$(2 z - y + 2) (2 z - y + 2)$

Step 2

Expand $(2 z - y + 2) (2 z - y + 2)$ by multiplying each term in the first expression by each term in the second expression.

$2 z (2 z) + 2 z (- y) + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3

Simplify each term.

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Step 3.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 z \cdot z + 2 z (- y) + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.2

Multiply $z$ by $z$ by adding the exponents.

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Step 3.2.1

Move $z$ .

$2 \cdot 2 (z \cdot z) + 2 z (- y) + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.2.2

Multiply $z$ by $z$ .

$2 \cdot 2 z^{2} + 2 z (- y) + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.3

Multiply $2$ by $2$ .

$4 z^{2} + 2 z (- y) + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.4

Rewrite using the commutative property of multiplication.

$4 z^{2} + 2 \cdot - 1 z y + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.5

Multiply $2$ by $- 1$ .

$4 z^{2} - 2 z y + 2 z \cdot 2 - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.6

Multiply $2$ by $2$ .

$4 z^{2} - 2 z y + 4 z - y (2 z) - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.7

Rewrite using the commutative property of multiplication.

$4 z^{2} - 2 z y + 4 z - 1 \cdot 2 y z - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.8

Multiply $- 1$ by $2$ .

$4 z^{2} - 2 z y + 4 z - 2 y z - y (- y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.9

Rewrite using the commutative property of multiplication.

$4 z^{2} - 2 z y + 4 z - 2 y z - 1 \cdot - 1 y \cdot y - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.10

Multiply $y$ by $y$ by adding the exponents.

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Step 3.10.1

Move $y$ .

$4 z^{2} - 2 z y + 4 z - 2 y z - 1 \cdot - 1 (y \cdot y) - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.10.2

Multiply $y$ by $y$ .

$4 z^{2} - 2 z y + 4 z - 2 y z - 1 \cdot - 1 y^{2} - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.11

Multiply $- 1$ by $- 1$ .

$4 z^{2} - 2 z y + 4 z - 2 y z + 1 y^{2} - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.12

Multiply $y^{2}$ by $1$ .

$4 z^{2} - 2 z y + 4 z - 2 y z + y^{2} - y \cdot 2 + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.13

Multiply $2$ by $- 1$ .

$4 z^{2} - 2 z y + 4 z - 2 y z + y^{2} - 2 y + 2 (2 z) + 2 (- y) + 2 \cdot 2$

Step 3.14

Multiply $2$ by $2$ .

$4 z^{2} - 2 z y + 4 z - 2 y z + y^{2} - 2 y + 4 z + 2 (- y) + 2 \cdot 2$

Step 3.15

Multiply $- 1$ by $2$ .

$4 z^{2} - 2 z y + 4 z - 2 y z + y^{2} - 2 y + 4 z - 2 y + 2 \cdot 2$

Step 3.16

Multiply $2$ by $2$ .

$4 z^{2} - 2 z y + 4 z - 2 y z + y^{2} - 2 y + 4 z - 2 y + 4$

Step 4

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

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Step 4.1

Move $z$ .

$4 z^{2} - 2 y z - 2 y z + 4 z + y^{2} - 2 y + 4 z - 2 y + 4$

Step 4.2

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

$4 z^{2} - 4 y z + 4 z + y^{2} - 2 y + 4 z - 2 y + 4$

Step 5

Add $4 z$ and $4 z$ .

$4 z^{2} - 4 y z + y^{2} - 2 y + 8 z - 2 y + 4$

Step 6

Subtract $2 y$ from $- 2 y$ .

$4 z^{2} - 4 y z + y^{2} - 4 y + 8 z + 4$

Step 7

Simplify the expression.

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Step 7.1

Move $4 z^{2}$ .

$- 4 y z + y^{2} + 4 z^{2} - 4 y + 8 z + 4$

Step 7.2

Reorder $- 4 y z$ and $y^{2}$ .

$y^{2} - 4 y z + 4 z^{2} - 4 y + 8 z + 4$