Solve x = 2 + sqrt(5); x ^ 2 + 1/(x ^ 2) | 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

$$x=2+\sqrt{5}; x^{2}+\frac{1}{x^{2}}$$

Solve for x, y

$y=18$

Solution Steps

Consider the second equation. Insert the known values of variables into the equation.

$$y=\left(2+\sqrt{5}\right)^{2}+\frac{1}{\left(2+\sqrt{5}\right)^{2}}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(2+\sqrt{5}\right)^{2}$.

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

The square of $\sqrt{5}$ is $5$.

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

Add $4$ and $5$ to get $9$.

$$y=9+4\sqrt{5}+\frac{1}{\left(2+\sqrt{5}\right)^{2}}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(2+\sqrt{5}\right)^{2}$.

$$y=9+4\sqrt{5}+\frac{1}{4+4\sqrt{5}+\left(\sqrt{5}\right)^{2}}$$

The square of $\sqrt{5}$ is $5$.

$$y=9+4\sqrt{5}+\frac{1}{4+4\sqrt{5}+5}$$

Add $4$ and $5$ to get $9$.

$$y=9+4\sqrt{5}+\frac{1}{9+4\sqrt{5}}$$

Rationalize the denominator of $\frac{1}{9+4\sqrt{5}}$ by multiplying numerator and denominator by $9-4\sqrt{5}$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)}$$

Consider $\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{9^{2}-\left(4\sqrt{5}\right)^{2}}$$

Calculate $9$ to the power of $2$ and get $81$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-\left(4\sqrt{5}\right)^{2}}$$

Expand $\left(4\sqrt{5}\right)^{2}$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-4^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate $4$ to the power of $2$ and get $16$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-16\left(\sqrt{5}\right)^{2}}$$

The square of $\sqrt{5}$ is $5$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-16\times 5}$$

Multiply $16$ and $5$ to get $80$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-80}$$

Subtract $80$ from $81$ to get $1$.

$$y=9+4\sqrt{5}+\frac{9-4\sqrt{5}}{1}$$

Anything divided by one gives itself.

$$y=9+4\sqrt{5}+9-4\sqrt{5}$$

Add $9$ and $9$ to get $18$.

$$y=18+4\sqrt{5}-4\sqrt{5}$$

Combine $4\sqrt{5}$ and $-4\sqrt{5}$ to get $0$.

$$y=18$$

The system is now solved.

$$x=2+\sqrt{5}$$ $$y=18$$