If `alpha` and `beta` are the roots of the quadratic equation `ax^2+bx+c=0`
then the sum of the roots `= alpha+beta=-b/a`
and the product of the roots `= alpha*beta=c/a`
Example :
1. Find the quadratic equation whose roots are Alpha = 3, Beta = -4
Solution:
Let `alpha=3` and `beta=-4`
Then, the sum of the roots `= alpha+beta=(3)+(-4)=-1`
and the product of the roots `= alpha*beta=(3)*(-4)=-12`
The equation with roots `alpha` and `beta` is given by
`x^2-(alpha+beta)x+alpha*beta=0`
`:.x^2-(-1)x+(-12)=0`
`:.x^2+x-12=0`
2. Find the quadratic equation whose roots are Alpha = -1/2, Beta = 2/3
Solution:
Let `alpha=-1/2` and `beta=2/3`
Then, the sum of the roots `= alpha+beta=(-1/2)+(2/3)=1/6`
and the product of the roots `= alpha*beta=(-1/2)*(2/3)=-1/3`
The equation with roots `alpha` and `beta` is given by
`x^2-(alpha+beta)x+alpha*beta=0`
`:.x^2-(1/6)x+(-1/3)=0`
`:.6x^2-x-2=0`
3. Find the quadratic equation whose roots are Alpha = 1+3sqrt(2), Beta = 1-3sqrt(2)
Solution:
Let `alpha=1+3sqrt(2)` and `beta=1-3sqrt(2)`
Then, the sum of the roots `= alpha+beta=(1+3sqrt(2))+(1-3sqrt(2))=2`
and the product of the roots `= alpha*beta=(1+3sqrt(2))*(1-3sqrt(2))=-17`
The equation with roots `alpha` and `beta` is given by
`x^2-(alpha+beta)x+alpha*beta=0`
`:.x^2-(2)x+(-17)=0`
`:.x^2-2x-17=0`
This material is intended as a summary. Use your textbook for detail explanation.
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