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Home > Algebra calculators > If `alpha` and `beta` are roots of quadratic equation, then find equation whose roots are `alpha^2` and `beta^2` example
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If `alpha` and `beta` are roots of quadratic equation `2x^2-3x-6=0`, then find equation whose roots are `alpha^2` and `beta^2` example
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1. Example-1
1. If `alpha` and `beta` are roots of quadratic equation `2x^2+3x-1=0`, then form the equation whose roots are `alpha/beta,beta/alpha`
Solution:
`2x^2+3x-1=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=2,b=3,c=-1`
Sum of roots `=alpha+beta=(-b)/a=(-3)/2`
Product of roots `=alpha*beta=c/a=(-1)/2`
Now, find equation whose roots are `alpha/beta,beta/alpha`
Sum of roots `=(alpha/beta)+(beta/alpha)`
`=(alpha^2+beta^2)/(betaalpha)`
`alpha^2+beta^2=13/4`
We know that
`alpha^2+beta^2=(alpha+beta)^2-2alphabeta`
`:.alpha^2+beta^2=((-3)/2)^2-2*(-1)/2`
`:.alpha^2+beta^2=9/4+1`
`:.alpha^2+beta^2=13/4`
`betaalpha=(-1)/2`
`:.(alpha^2+beta^2)/(betaalpha)=(13/4)/((-1)/2)=(-13)/2`
`:.` Sum of roots `=(-13)/2`
Product of roots `=(alpha/beta)*(beta/alpha)=1`
`:.` Required equation is
`x^2-((-13)/2)x+1=0`
`:.2x^2+13x+2=0`
2. If `alpha` and `beta` are roots of quadratic equation `2x^2+3x-1=0`, then form the equation whose roots are `alpha^2+2,beta^2+2`
Solution:
`2x^2+3x-1=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=2,b=3,c=-1`
Sum of roots `=alpha+beta=(-b)/a=(-3)/2`
Product of roots `=alpha*beta=c/a=(-1)/2`
Now, find equation whose roots are `alpha^2+2,beta^2+2`
Sum of roots `=(alpha^2+2)+(beta^2+2)=alpha^2+beta^2+4`
`alpha^2+beta^2=13/4``alpha^2+beta^2=13/4`
We know that
`alpha^2+beta^2=(alpha+beta)^2-2alphabeta`
`:.alpha^2+beta^2=((-3)/2)^2-2*(-1)/2`
`:.alpha^2+beta^2=9/4+1`
`:.alpha^2+beta^2=13/4`
`:.alpha^2+beta^2=13/4`
`:.alpha^2+beta^2+4=13/4+4=29/4`
`:.` Sum of roots `=29/4`
Product of roots `=(alpha^2+2)*(beta^2+2)=alpha^2beta^2+2alpha^2+2beta^2+4`
`2alpha^2+2beta^2=13/2``alpha^2+beta^2=13/4`
We know that
`alpha^2+beta^2=(alpha+beta)^2-2alphabeta`
`:.alpha^2+beta^2=((-3)/2)^2-2*(-1)/2`
`:.alpha^2+beta^2=9/4+1`
`:.alpha^2+beta^2=13/4`
`:.2*(alpha^2+beta^2)=2xx13/4=13/2` `:.2alpha^2+2beta^2=13/2`
`alpha^2beta^2=1/4``:.alpha^2beta^2=1/4`
`:.alpha^2beta^2+2alpha^2+2beta^2+4=13/2+1/4+4=43/4`
`:.` Product of roots `=43/4`
`:.` Required equation is
`x^2-29/4x+43/4=0`
`:.4x^2-29x+43=0`
3. If `alpha` and `beta` are roots of quadratic equation `2x^2+3x-1=0`, then form the equation whose roots are `alpha^2beta,beta^2alpha`
Solution:
`2x^2+3x-1=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=2,b=3,c=-1`
Sum of roots `=alpha+beta=(-b)/a=(-3)/2`
Product of roots `=alpha*beta=c/a=(-1)/2`
Now, find equation whose roots are `alpha^2beta,beta^2alpha`
Sum of roots `=alpha^2beta+beta^2alpha`
`=alphabeta(alpha+beta)`
`:.alphabeta(alpha+beta)=(-1)/2xx(-3)/2=3/4`
`:.` Sum of roots `=3/4`
Product of roots `=alpha^2beta*beta^2alpha=alpha^3beta^3`
`:.` Product of roots `=(-1)/8`
`:.` Required equation is
`x^2-3/4x+((-1)/8)=0`
`:.8x^2-6x-1=0`
4. If `alpha` and `beta` are roots of quadratic equation `2x^2+3x-1=0`, then form the equation whose roots are `2alpha+3beta,3alpha+2beta`
Solution:
`2x^2+3x-1=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=2,b=3,c=-1`
Sum of roots `=alpha+beta=(-b)/a=(-3)/2`
Product of roots `=alpha*beta=c/a=(-1)/2`
Now, find equation whose roots are `2alpha+3beta,3alpha+2beta`
Sum of roots `=(2alpha+3beta)+(3alpha+2beta)=5alpha+5beta`
`=5*(alpha+beta)`
`:.5*(alpha+beta)=5xx(-3)/2=(-15)/2`
`:.` Sum of roots `=(-15)/2`
Product of roots `=(2alpha+3beta)*(3alpha+2beta)=6alpha^2+4alphabeta+9betaalpha+6beta^2=6alpha^2+13alphabeta+6beta^2`
`6alpha^2+6beta^2=39/2``alpha^2+beta^2=13/4`
We know that
`alpha^2+beta^2=(alpha+beta)^2-2alphabeta`
`:.alpha^2+beta^2=((-3)/2)^2-2*(-1)/2`
`:.alpha^2+beta^2=9/4+1`
`:.alpha^2+beta^2=13/4`
`:.6*(alpha^2+beta^2)=6xx13/4=39/2` `:.6alpha^2+6beta^2=39/2`
`13alphabeta=(-13)/2``:.13alphabeta=(-13)/2`
`:.6alpha^2+13alphabeta+6beta^2=39/2-13/2=13`
`:.` Product of roots `=13`
`:.` Required equation is
`x^2-((-15)/2)x+13=0`
`:.2x^2+15x+26=0`
This material is intended as a summary. Use your textbook for detail explanation.
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