Simplify (-4+ square root of 3)^2

${(- 4 + \sqrt{3})}^{2}$

Step 1

Rewrite ${(- 4 + \sqrt{3})}^{2}$ as $(- 4 + \sqrt{3}) (- 4 + \sqrt{3})$ .

$(- 4 + \sqrt{3}) (- 4 + \sqrt{3})$

Step 2

Expand $(- 4 + \sqrt{3}) (- 4 + \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$- 4 (- 4 + \sqrt{3}) + \sqrt{3} (- 4 + \sqrt{3})$

Apply the distributive property.

$- 4 \cdot - 4 - 4 \sqrt{3} + \sqrt{3} (- 4 + \sqrt{3})$

Apply the distributive property.

$- 4 \cdot - 4 - 4 \sqrt{3} + \sqrt{3} \cdot - 4 + \sqrt{3} \sqrt{3}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $- 4$ by $- 4$ .

$16 - 4 \sqrt{3} + \sqrt{3} \cdot - 4 + \sqrt{3} \sqrt{3}$

Move $- 4$ to the left of $\sqrt{3}$ .

$16 - 4 \sqrt{3} - 4 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}$

Combine using the product rule for radicals.

$16 - 4 \sqrt{3} - 4 \sqrt{3} + \sqrt{3 \cdot 3}$

Multiply $3$ by $3$ .

$16 - 4 \sqrt{3} - 4 \sqrt{3} + \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$16 - 4 \sqrt{3} - 4 \sqrt{3} + \sqrt{3^{2}}$

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

$16 - 4 \sqrt{3} - 4 \sqrt{3} + 3$

Add $16$ and $3$ .

$19 - 4 \sqrt{3} - 4 \sqrt{3}$

Subtract $4 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$19 - 8 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$19 - 8 \sqrt{3}$

Decimal Form:

$5.14359353 \dots$