Solve Matrix

$$\left. \begin{array} { l } { \text { If } a + 1 / a = 4 \text { then the value of } a ^ { 2 } + } \\ { \left. \begin{array} { l l } { \text { (a) } 14 } & { \text { (b) } 16 } & { \text { (c) } 18 } \\ { \text { If } x ^ { 2 } + 1 / x ^ { 2 } = 6 } & { \text { then } x - 1 / x = } \end{array} \right. } \end{array} \right.$$

Solve for a

$$a=\sqrt{3}+2\approx 3.732050808$$

$$a=2-\sqrt{3}\approx 0.267949192$$

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.

$$aa+1=4a$$

Multiply a and a to get a^{2}.

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

Subtract 4a from both sides.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4}}{2}$$

Square -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4}}{2}$$

Add 16 to -4.

$$a=\frac{-\left(-4\right)±\sqrt{12}}{2}$$

Take the square root of 12.

$$a=\frac{-\left(-4\right)±2\sqrt{3}}{2}$$

The opposite of -4 is 4.

$$a=\frac{4±2\sqrt{3}}{2}$$

Now solve the equation a=\frac{4±2\sqrt{3}}{2} when ± is plus. Add 4 to 2\sqrt{3}.

$$a=\frac{2\sqrt{3}+4}{2}$$

Divide 4+2\sqrt{3} by 2.

$$a=\sqrt{3}+2$$

Now solve the equation a=\frac{4±2\sqrt{3}}{2} when ± is minus. Subtract 2\sqrt{3} from 4.

$$a=\frac{4-2\sqrt{3}}{2}$$

Divide 4-2\sqrt{3} by 2.

$$a=2-\sqrt{3}$$

The equation is now solved.

$$a=\sqrt{3}+2 a=2-\sqrt{3}$$

Steps for Completing the Square

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

$$aa+1=4a$$

Multiply a and a to get a^{2}.

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

Subtract 4a from both sides.

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

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$a^{2}-4a=-1$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-4a+\left(-2\right)^{2}=-1+\left(-2\right)^{2}$$

Square -2.

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

Add -1 to 4.

$$a^{2}-4a+4=3$$

Factor a^{2}-4a+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-2\right)^{2}=3$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{3}$$

Simplify.

$$a-2=\sqrt{3} a-2=-\sqrt{3}$$

Add 2 to both sides of the equation.

$$a=\sqrt{3}+2 a=2-\sqrt{3}$$