Solve {l}{x={(4-sqrt{15})}}{y={(x^2+1/x^2)}}{z=y}{a=z}{b=a}{text{Solvefor}ctext{where}}{c=b}

$$\left. \begin{array} { l } { x = {(4 - \sqrt{15})} }\\ { y = {(x ^ {2} + \frac{1}{x ^ {2}})} }\\ { z = y }\\ { a = z }\\ { b = a }\\ { \text{Solve for } c \text{ where} } \\ { c = b } \end{array} \right.$$

Solve for x, y, z, a, b, c

$$c=62$$

Solution Steps

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

$$y=\left(4-\sqrt{15}\right)^{2}+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{15}\right)^{2}.

$$y=16-8\sqrt{15}+\left(\sqrt{15}\right)^{2}+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$y=16-8\sqrt{15}+15+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Add 16 and 15 to get 31.

$$y=31-8\sqrt{15}+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{15}\right)^{2}.

$$y=31-8\sqrt{15}+\frac{1}{16-8\sqrt{15}+\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$y=31-8\sqrt{15}+\frac{1}{16-8\sqrt{15}+15}$$

Add 16 and 15 to get 31.

$$y=31-8\sqrt{15}+\frac{1}{31-8\sqrt{15}}$$

Rationalize the denominator of \frac{1}{31-8\sqrt{15}} by multiplying numerator and denominator by 31+8\sqrt{15}.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{\left(31-8\sqrt{15}\right)\left(31+8\sqrt{15}\right)}$$

Consider \left(31-8\sqrt{15}\right)\left(31+8\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{31^{2}-\left(-8\sqrt{15}\right)^{2}}$$

Calculate 31 to the power of 2 and get 961.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-\left(-8\sqrt{15}\right)^{2}}$$

Expand \left(-8\sqrt{15}\right)^{2}.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-\left(-8\right)^{2}\left(\sqrt{15}\right)^{2}}$$

Calculate -8 to the power of 2 and get 64.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-64\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-64\times 15}$$

Multiply 64 and 15 to get 960.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-960}$$

Subtract 960 from 961 to get 1.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{1}$$

Anything divided by one gives itself.

$$y=31-8\sqrt{15}+31+8\sqrt{15}$$

Add 31 and 31 to get 62.

$$y=62-8\sqrt{15}+8\sqrt{15}$$

Combine -8\sqrt{15} and 8\sqrt{15} to get 0.

$$y=62$$

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

$$z=62$$

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

$$a=62$$

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

$$b=62$$

Consider the equation (6). Insert the known values of variables into the equation.

$$c=62$$

The system is now solved.

$$x=4-\sqrt{15} y=62 z=62 a=62 b=62 c=62$$