Solve Matrix

$$\left. \begin{array} { l } { \text { If } ( a ^ { 2 } + \frac { 1 } { a ^ { 2 } } ) = 47 \text { and } a \neq 0 ; \text { find: } } \\ { \left. \begin{array} { l l } { \text { a. } ( a + \frac { 1 } { a } ) } & { \text { b. } ( a ^ { 3 } + \frac { 1 } { a ^ { 3 } } ) } \end{array} \right. } \end{array} \right.$$

Solve for a

$$a=\frac{-3\sqrt{5}-7}{2}\approx -6.854101966$$

$$a = \frac{3 \sqrt{5} + 7}{2} \approx 6.854101966$$

$$a=\frac{7-3\sqrt{5}}{2}\approx 0.145898034$$

$$a=\frac{3\sqrt{5}-7}{2}\approx -0.145898034$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a^{2}.

$$a^{2}a^{2}+1=47a^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$a^{4}+1=47a^{2}$$

Subtract 47a^{2} from both sides.

$$a^{4}+1-47a^{2}=0$$

Substitute t for a^{2}.

$$t^{2}-47t+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -47 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-47\right)±\sqrt{\left(-47\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{47±21\sqrt{5}}{2}$$

Solve the equation t=\frac{47±21\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=\frac{21\sqrt{5}+47}{2} t=\frac{47-21\sqrt{5}}{2}$$

Since a=t^{2}, the solutions are obtained by evaluating a=±\sqrt{t} for each t.

$$a=\frac{3\sqrt{5}+7}{2} a=-\frac{3\sqrt{5}+7}{2} a=\frac{7-3\sqrt{5}}{2} a=-\frac{7-3\sqrt{5}}{2}$$