Solve 4/(3 - sqrt(5)) + 1/(2 - sqrt(5)) + (3sqrt(5))/4 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{4}{3-\sqrt{5}}+\frac{1}{2-\sqrt{5}}+\frac{3\sqrt{5}}{4}$$

Evaluate

$\frac{3\sqrt{5}}{4}+1\approx 2.677050983$

Short Solution Steps

Rationalize the denominator of $\frac{4}{3-\sqrt{5}}$ by multiplying numerator and denominator by $3+\sqrt{5}$.

$$\frac{4\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}+\frac{1}{2-\sqrt{5}}+\frac{3\sqrt{5}}{4}$$

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

$$\frac{4\left(3+\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}+\frac{1}{2-\sqrt{5}}+\frac{3\sqrt{5}}{4}$$

Square $3$. Square $\sqrt{5}$.

$$\frac{4\left(3+\sqrt{5}\right)}{9-5}+\frac{1}{2-\sqrt{5}}+\frac{3\sqrt{5}}{4}$$

Subtract $5$ from $9$ to get $4$.

$$\frac{4\left(3+\sqrt{5}\right)}{4}+\frac{1}{2-\sqrt{5}}+\frac{3\sqrt{5}}{4}$$

Cancel out $4$ and $4$.

$$3+\sqrt{5}+\frac{1}{2-\sqrt{5}}+\frac{3\sqrt{5}}{4}$$

Rationalize the denominator of $\frac{1}{2-\sqrt{5}}$ by multiplying numerator and denominator by $2+\sqrt{5}$.

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

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

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

Square $2$. Square $\sqrt{5}$.

$$3+\sqrt{5}+\frac{2+\sqrt{5}}{4-5}+\frac{3\sqrt{5}}{4}$$

Subtract $5$ from $4$ to get $-1$.

$$3+\sqrt{5}+\frac{2+\sqrt{5}}{-1}+\frac{3\sqrt{5}}{4}$$

Anything divided by -1 gives its opposite. To find the opposite of $2+\sqrt{5}$, find the opposite of each term.

$$3+\sqrt{5}-2-\sqrt{5}+\frac{3\sqrt{5}}{4}$$

Subtract $2$ from $3$ to get $1$.

$$1+\sqrt{5}-\sqrt{5}+\frac{3\sqrt{5}}{4}$$

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

$$1+\frac{3\sqrt{5}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $1$ times $\frac{4}{4}$.

$$\frac{4}{4}+\frac{3\sqrt{5}}{4}$$

Since $\frac{4}{4}$ and $\frac{3\sqrt{5}}{4}$ have the same denominator, add them by adding their numerators.

$$\frac{4+3\sqrt{5}}{4}$$

Factor

$\frac{3 \sqrt{5} + 4}{4} = 2.6770509831248424$