If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f(x)=x^{2}+x-2$, find the value of $\frac{1}{\alpha}-\frac{1}{\beta}$.
Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomials $f(x)=x^{2}+x-2$
Sum of the zeros $=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}$
$\alpha+\beta=-\left[\frac{1}{1}\right]$
$\alpha+\beta=-\frac{1}{1}$
$\alpha+\beta=-1$
Product if zeros $=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}$
$\alpha \beta=\frac{-2}{1}$
$\alpha \beta=-2$
We have, $\frac{1}{\alpha}-\frac{1}{\beta}$
$\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^{2}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}+\frac{2}{\alpha \beta}$
$\left(\frac{\alpha+\beta}{\alpha \beta}\right)^{2}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}+\frac{2}{\alpha \beta}$
$\left(\frac{\alpha+\beta}{\alpha \beta}\right)^{2}-\frac{2}{\alpha \beta}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$
By substituting $\alpha+\beta=-1$ and $\alpha \beta=-2$ we get,
$\left(\frac{-1}{-2}\right)^{2}-\frac{2}{-2}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$
$\frac{1}{4}+1=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$
$\frac{1}{4}+\frac{1 \times 4}{1 \times 4}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$
$\frac{1+4}{4}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$
$\frac{5}{4}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$
$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}-\frac{2}{\alpha \beta}$
By substituting $\frac{5}{4}=\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}$ in $\left(\frac{1}{\alpha}\right)^{2}+\left(\frac{1}{\beta}\right)^{2}-\frac{2}{\alpha \beta}$ we get,
$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}=\frac{5}{4}-\frac{2}{-2}$
$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}=\frac{5}{4}+1$
$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}=\frac{5}{4}+\frac{1 \times 4}{1 \times 4}$
$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}=\frac{5+4}{4}$
$\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}=\frac{9}{4}$
Taking square root on both sides we get
$\sqrt{\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)^{2}}=\sqrt{\frac{3 \times 3}{2 \times 2}}$
$\frac{1}{\alpha}-\frac{1}{\beta}=\pm \frac{3}{2}$
Hence, the value of $\frac{1}{\alpha}-\frac{1}{\beta}$ is $\pm \frac{3}{2}$.
- 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
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
All Study Material
- 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
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,