Question:
If $x+\frac{1}{x}=\sqrt{5}$, find the value of $x^{2}+\frac{1}{x^{2}}$ and $x^{4}+\frac{1}{x^{4}}$
Solution:
We have,
$(x+1 / x)^{2}=x^{2}+(1 / x)^{2}+2 * x * 1 / x$
$\Rightarrow(x+1 / x)^{2}=x^{2}+1 / x^{2}+2$
$\Rightarrow(\sqrt{5})^{2}=x^{2}+\frac{1}{x^{2}}+2\left[\therefore x+\frac{1}{x}=\sqrt{5}\right]$
$\Rightarrow 5=x^{2}+1 / x^{2}+2$
$\Rightarrow x^{2}+1 / x^{2}=3 \ldots(1)$
Now, $\left(x^{2}+1 / x^{2}\right)^{2}=x^{4}+1 / x^{4}+2 * x^{2} * 1 / x^{2}$
$\Rightarrow\left(x^{2}+1 \times 2\right)^{2}=x^{4}+1 / x^{4}+2$
$\Rightarrow 9=x^{4}+1 / x^{4}+2\left[\therefore x^{2}+1 / x^{2}=3\right]$
$\Rightarrow x^{4}+1 / x^{4}=7$
Hence, $x^{2}+1 / x^{2}=3 ; x^{4}+1 / x^{4}=7 .$
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- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
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- Atoms
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