Solve 12/3-sqrt{5}-3/2+sqrt{5}

$$\frac{ 12 }{ 3- \sqrt{ 5 } } - \frac{ 3 }{ 2+ \sqrt{ 5 } }$$

Evaluate

$$15$$

Solution Steps

Rationalize the denominator of \frac{12}{3-\sqrt{5}} by multiplying numerator and denominator by 3+\sqrt{5}.

$$\frac{12\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}-\frac{3}{2+\sqrt{5}}$$

Consider \left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{12\left(3+\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}-\frac{3}{2+\sqrt{5}}$$

Square 3. Square \sqrt{5}.

$$\frac{12\left(3+\sqrt{5}\right)}{9-5}-\frac{3}{2+\sqrt{5}}$$

Subtract 5 from 9 to get 4.

$$\frac{12\left(3+\sqrt{5}\right)}{4}-\frac{3}{2+\sqrt{5}}$$

Divide 12\left(3+\sqrt{5}\right) by 4 to get 3\left(3+\sqrt{5}\right).

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

Rationalize the denominator of \frac{3}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

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

Consider \left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

$$3\left(3+\sqrt{5}\right)-\frac{3\left(2-\sqrt{5}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

The opposite of -3\left(2-\sqrt{5}\right) is 3\left(2-\sqrt{5}\right).

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

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

$$9+3\sqrt{5}+3\left(2-\sqrt{5}\right)$$

Use the distributive property to multiply 3 by 2-\sqrt{5}.

$$9+3\sqrt{5}+6-3\sqrt{5}$$

Add 9 and 6 to get 15.

$$15+3\sqrt{5}-3\sqrt{5}$$

Combine 3\sqrt{5} and -3\sqrt{5} to get 0.

$$15$$

Factor

$$3\times 5$$