Simplify (2+ square root of 5)^2

${(2 + \sqrt{5})}^{2}$

Step 1

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

$(2 + \sqrt{5}) (2 + \sqrt{5})$

Step 2

Expand $(2 + \sqrt{5}) (2 + \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$2 (2 + \sqrt{5}) + \sqrt{5} (2 + \sqrt{5})$

Apply the distributive property.

$2 \cdot 2 + 2 \sqrt{5} + \sqrt{5} (2 + \sqrt{5})$

Apply the distributive property.

$2 \cdot 2 + 2 \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \sqrt{5}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$4 + 2 \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \sqrt{5}$

Move $2$ to the left of $\sqrt{5}$ .

$4 + 2 \sqrt{5} + 2 \cdot \sqrt{5} + \sqrt{5} \sqrt{5}$

Combine using the product rule for radicals.

$4 + 2 \sqrt{5} + 2 \sqrt{5} + \sqrt{5 \cdot 5}$

Multiply $5$ by $5$ .

$4 + 2 \sqrt{5} + 2 \sqrt{5} + \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

$4 + 2 \sqrt{5} + 2 \sqrt{5} + \sqrt{5^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$4 + 2 \sqrt{5} + 2 \sqrt{5} + 5$

Add $4$ and $5$ .

$9 + 2 \sqrt{5} + 2 \sqrt{5}$

Add $2 \sqrt{5}$ and $2 \sqrt{5}$ .

$9 + 4 \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$9 + 4 \sqrt{5}$

Decimal Form:

$17.94427190 \dots$