Solve (a+1/a)^2=3.text{Nilaidari}(a^3+1/a^3)=

$$( a + \frac { 1 } { a } ) ^ { 2 } = 3 . \text { Nilai dari } ( a ^ { 3 } + \frac { 1 } { a ^ { 3 } } ) =$$

Solve for a

$$a=-\frac{\sqrt{3}}{2}-\frac{1}{2}i\approx -0.866025404-0.5i$$

$$a=-\frac{\sqrt{3}}{2}+\frac{1}{2}i\approx -0.866025404+0.5i$$

$$a=\frac{\sqrt{3}}{2}+\frac{1}{2}i\approx 0.866025404+0.5i$$

$$a=\frac{\sqrt{3}}{2}-\frac{1}{2}i\approx 0.866025404-0.5i$$

Steps Using the Quadratic Formula

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

$$\left(\frac{aa}{a}+\frac{1}{a}\right)^{2}=3$$

Since \frac{aa}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$\left(\frac{aa+1}{a}\right)^{2}=3$$

Do the multiplications in aa+1.

$$\left(\frac{a^{2}+1}{a}\right)^{2}=3$$

To raise \frac{a^{2}+1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(a^{2}+1\right)^{2}}{a^{2}}=3$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a^{2}+1\right)^{2}.

$$\frac{\left(a^{2}\right)^{2}+2a^{2}+1}{a^{2}}=3$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{a^{4}+2a^{2}+1}{a^{2}}=3$$

Subtract 3 from both sides.

$$\frac{a^{4}+2a^{2}+1}{a^{2}}-3=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{a^{2}}{a^{2}}.

$$\frac{a^{4}+2a^{2}+1}{a^{2}}-\frac{3a^{2}}{a^{2}}=0$$

Since \frac{a^{4}+2a^{2}+1}{a^{2}} and \frac{3a^{2}}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a^{4}+2a^{2}+1-3a^{2}}{a^{2}}=0$$

Combine like terms in a^{4}+2a^{2}+1-3a^{2}.

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

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

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

Substitute t for a^{2}.

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

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, -1 for b, and 1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{1±\sqrt{-3}}{2}$$

Solve the equation t=\frac{1±\sqrt{-3}}{2} when ± is plus and when ± is minus.

$$t=\frac{1+\sqrt{3}i}{2} t=\frac{-\sqrt{3}i+1}{2}$$

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

$$a=-e^{\frac{\pi i}{6}} a=e^{\frac{\pi i}{6}} a=-ie^{\frac{\pi i}{3}} a=ie^{\frac{\pi i}{3}}$$

Variable a cannot be equal to 0.

$$a=ie^{\frac{\pi i}{3}}\text{, }a\neq 0 a=-ie^{\frac{\pi i}{3}}\text{, }a\neq 0 a=e^{\frac{\pi i}{6}}\text{, }a\neq 0 a=-e^{\frac{\pi i}{6}}\text{, }a\neq 0$$