Basic Math Examples

${(2 z + 5)}^{2} - {(z - 3)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(2 z + 5)}^{2}$ as $(2 z + 5) (2 z + 5)$ .

$(2 z + 5) (2 z + 5) - {(z - 3)}^{2}$

Step 1.2

Expand $(2 z + 5) (2 z + 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$2 z (2 z + 5) + 5 (2 z + 5) - {(z - 3)}^{2}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $z$ .

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

Step 1.3.1.2.2

Multiply $z$ by $z$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

$4 z^{2} + 2 z \cdot 5 + 5 (2 z) + 5 \cdot 5 - {(z - 3)}^{2}$

Step 1.3.1.4

Multiply $5$ by $2$ .

$4 z^{2} + 10 z + 5 (2 z) + 5 \cdot 5 - {(z - 3)}^{2}$

Step 1.3.1.5

Multiply $2$ by $5$ .

$4 z^{2} + 10 z + 10 z + 5 \cdot 5 - {(z - 3)}^{2}$

Step 1.3.1.6

Multiply $5$ by $5$ .

$4 z^{2} + 10 z + 10 z + 25 - {(z - 3)}^{2}$

Step 1.3.2

Add $10 z$ and $10 z$ .

$4 z^{2} + 20 z + 25 - {(z - 3)}^{2}$

Step 1.4

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

$4 z^{2} + 20 z + 25 - ((z - 3) (z - 3))$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$4 z^{2} + 20 z + 25 - (z (z - 3) - 3 (z - 3))$

Step 1.5.2

Apply the distributive property.

$4 z^{2} + 20 z + 25 - (z \cdot z + z \cdot - 3 - 3 (z - 3))$

Step 1.5.3

Apply the distributive property.

$4 z^{2} + 20 z + 25 - (z \cdot z + z \cdot - 3 - 3 z - 3 \cdot - 3)$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $z$ by $z$ .

$4 z^{2} + 20 z + 25 - (z^{2} + z \cdot - 3 - 3 z - 3 \cdot - 3)$

Step 1.6.1.2

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

$4 z^{2} + 20 z + 25 - (z^{2} - 3 \cdot z - 3 z - 3 \cdot - 3)$

Step 1.6.1.3

Multiply $- 3$ by $- 3$ .

$4 z^{2} + 20 z + 25 - (z^{2} - 3 z - 3 z + 9)$

Step 1.6.2

Subtract $3 z$ from $- 3 z$ .

$4 z^{2} + 20 z + 25 - (z^{2} - 6 z + 9)$

Step 1.7

Apply the distributive property.

$4 z^{2} + 20 z + 25 - z^{2} - (- 6 z) - 1 \cdot 9$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $- 6$ by $- 1$ .

$4 z^{2} + 20 z + 25 - z^{2} + 6 z - 1 \cdot 9$

Step 1.8.2

Multiply $- 1$ by $9$ .

$4 z^{2} + 20 z + 25 - z^{2} + 6 z - 9$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $z^{2}$ from $4 z^{2}$ .

$3 z^{2} + 20 z + 25 + 6 z - 9$

Step 2.2

Add $20 z$ and $6 z$ .

$3 z^{2} + 26 z + 25 - 9$

Step 2.3

Subtract $9$ from $25$ .

$3 z^{2} + 26 z + 16$