Solve frac{3{left(sqrt{3}-4right)}^2+5left(sqrt{3}-4right)+2}{2left(sqrt{3}-4right)}

$$\frac{ 3 { \left( \sqrt{ 3 } -4 \right) }^{ 2 } +5 \left( \sqrt{ 3 } -4 \right) +2 }{ 2 \left( \sqrt{ 3 } -4 \right) }$$

Evaluate

$$\frac{37\sqrt{3}-99}{26}\approx -1.342850774$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-4\right)^{2}.

$$\frac{3\left(\left(\sqrt{3}\right)^{2}-8\sqrt{3}+16\right)+5\left(\sqrt{3}-4\right)+2}{2\left(\sqrt{3}-4\right)}$$

The square of \sqrt{3} is 3.

$$\frac{3\left(3-8\sqrt{3}+16\right)+5\left(\sqrt{3}-4\right)+2}{2\left(\sqrt{3}-4\right)}$$

Add 3 and 16 to get 19.

$$\frac{3\left(19-8\sqrt{3}\right)+5\left(\sqrt{3}-4\right)+2}{2\left(\sqrt{3}-4\right)}$$

Use the distributive property to multiply 3 by 19-8\sqrt{3}.

$$\frac{57-24\sqrt{3}+5\left(\sqrt{3}-4\right)+2}{2\left(\sqrt{3}-4\right)}$$

Use the distributive property to multiply 5 by \sqrt{3}-4.

$$\frac{57-24\sqrt{3}+5\sqrt{3}-20+2}{2\left(\sqrt{3}-4\right)}$$

Combine -24\sqrt{3} and 5\sqrt{3} to get -19\sqrt{3}.

$$\frac{57-19\sqrt{3}-20+2}{2\left(\sqrt{3}-4\right)}$$

Subtract 20 from 57 to get 37.

$$\frac{37-19\sqrt{3}+2}{2\left(\sqrt{3}-4\right)}$$

Add 37 and 2 to get 39.

$$\frac{39-19\sqrt{3}}{2\left(\sqrt{3}-4\right)}$$

Use the distributive property to multiply 2 by \sqrt{3}-4.

$$\frac{39-19\sqrt{3}}{2\sqrt{3}-8}$$

Rationalize the denominator of \frac{39-19\sqrt{3}}{2\sqrt{3}-8} by multiplying numerator and denominator by 2\sqrt{3}+8.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{\left(2\sqrt{3}-8\right)\left(2\sqrt{3}+8\right)}$$

Consider \left(2\sqrt{3}-8\right)\left(2\sqrt{3}+8\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{\left(2\sqrt{3}\right)^{2}-8^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{2^{2}\left(\sqrt{3}\right)^{2}-8^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{4\left(\sqrt{3}\right)^{2}-8^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{4\times 3-8^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{12-8^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{12-64}$$

Subtract 64 from 12 to get -52.

$$\frac{\left(39-19\sqrt{3}\right)\left(2\sqrt{3}+8\right)}{-52}$$

Use the distributive property to multiply 39-19\sqrt{3} by 2\sqrt{3}+8 and combine like terms.

$$\frac{-74\sqrt{3}+312-38\left(\sqrt{3}\right)^{2}}{-52}$$

The square of \sqrt{3} is 3.

$$\frac{-74\sqrt{3}+312-38\times 3}{-52}$$

Multiply -38 and 3 to get -114.

$$\frac{-74\sqrt{3}+312-114}{-52}$$

Subtract 114 from 312 to get 198.

$$\frac{-74\sqrt{3}+198}{-52}$$