Solve frac{sqrt{3}-2}{sqrt{3}+2}+(1-2sqrt{3})^2-1

$$\frac { \sqrt { 3 } - 2 } { \sqrt { 3 } + 2 } + ( 1 - 2 \sqrt { 3 } ) ^ { 2 } - 1$$

Evaluate

$$5$$

Solution Steps

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

$$\frac{\left(\sqrt{3}-2\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{3}+2\right)\left(\sqrt{3}-2\right)}+\left(1-2\sqrt{3}\right)^{2}-1$$

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

$$\frac{\left(\sqrt{3}-2\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{3}\right)^{2}-2^{2}}+\left(1-2\sqrt{3}\right)^{2}-1$$

Square \sqrt{3}. Square 2.

$$\frac{\left(\sqrt{3}-2\right)\left(\sqrt{3}-2\right)}{3-4}+\left(1-2\sqrt{3}\right)^{2}-1$$

Subtract 4 from 3 to get -1.

$$\frac{\left(\sqrt{3}-2\right)\left(\sqrt{3}-2\right)}{-1}+\left(1-2\sqrt{3}\right)^{2}-1$$

Multiply \sqrt{3}-2 and \sqrt{3}-2 to get \left(\sqrt{3}-2\right)^{2}.

$$\frac{\left(\sqrt{3}-2\right)^{2}}{-1}+\left(1-2\sqrt{3}\right)^{2}-1$$

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

$$\frac{\left(\sqrt{3}\right)^{2}-4\sqrt{3}+4}{-1}+\left(1-2\sqrt{3}\right)^{2}-1$$

The square of \sqrt{3} is 3.

$$\frac{3-4\sqrt{3}+4}{-1}+\left(1-2\sqrt{3}\right)^{2}-1$$

Add 3 and 4 to get 7.

$$\frac{7-4\sqrt{3}}{-1}+\left(1-2\sqrt{3}\right)^{2}-1$$

Anything divided by -1 gives its opposite. To find the opposite of 7-4\sqrt{3}, find the opposite of each term.

$$-7+4\sqrt{3}+\left(1-2\sqrt{3}\right)^{2}-1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-2\sqrt{3}\right)^{2}.

$$-7+4\sqrt{3}+1-4\sqrt{3}+4\left(\sqrt{3}\right)^{2}-1$$

The square of \sqrt{3} is 3.

$$-7+4\sqrt{3}+1-4\sqrt{3}+4\times 3-1$$

Multiply 4 and 3 to get 12.

$$-7+4\sqrt{3}+1-4\sqrt{3}+12-1$$

Add 1 and 12 to get 13.

$$-7+4\sqrt{3}+13-4\sqrt{3}-1$$

Add -7 and 13 to get 6.

$$6+4\sqrt{3}-4\sqrt{3}-1$$

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

$$6-1$$

Subtract 1 from 6 to get 5.

$$5$$

Factor

$$5$$