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प्रश्न
`12abx^2-(9a^2-8b^2)x-6ab=0,` `Where a≠0 and b≠0`
उत्तर
Given:
`12abx^2-(9a^2-8b^2)x-6ab=0,`
On comparing it with `Ax^2+Bx+C=0`, We get
`A=12ab, B=-(9a^2-8b^2)x-6ab=0`
Discriminant D is given by:
`D=B^2-4AC`
=`[-(9a^2-8b^2)]^2-4xx12abxx(-6ab)`
=`81a^4-144a^2b^2+64b^4+288a^2b^2`
=`81a^4+144a^2b^2+64b^2`
=`(9a^2+8b^2)^2>0`
Hence, the roots of the equation are equal.
Roots α and β are given by:
α=`(-B+sqrt(D))/(2A)=-([-9a^2-8b^2]+(sqrt(9a^2+8b^2)^2))/(2xx12ab)=(9a^2-8b^2+9a^2+8b^2)/(24ab)=(18a^2)/(24ab)=(3a)/(4b)`
`β=(-B-sqrt(D))/(2A)=-([-9a^2+8b^2]-(sqrt(9a^2-8b^2)^2))/(2xx12ab)=(9a^2-8b^2-9a^2-8b^2)/(24ab)=(-16a^2)/(24ab)=(-2b)/(3a)`
Thus, the roots of the equation are `(3a)/(4b)` and `(-2b)/(3a)`
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