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Solve the following quations by using qardratic formula: `abx^(2)+(b^(2)-ac)x-bc=0`

Answer» The given equation is `abx^(2)+(b^(2)-ac)x-bc=0.`
This is of the form `Ax^(2)+Bx+C=0,` where
`A=ab,B=(b^(2)-ac)" and "C=-bc.`
`:." "D=(B^(2)-4AC)=(b^(2)-ac)^(2)+4ab^(2)c`
`=b^(4)+a^(2)c^(2)-2ab^(2)c+4ab^(2)c`
`=b^(4)+a^(2)c^(2)+2ab^(2)c=(b^(2)+ac)^(2)gt0..`
So, the given equation has real roots.
Now, `sqrt(D)=(b^(2)+ac).`
`:." "alpha=(-B+sqrt(D))/(2A)=(-(b^(2)-ac)+(b^(2)+ac))/(2ab)=(2ac)/(2ab)=(c)/(b),`
`" "beta=(-B-sqrt(D))/(2A)=(-(b^(2)-ac)+(b^(2)+ac))/(2ab)=(-2b^(2))/(2ab)=(-b)/(a).`
Hence,`(c)/(b)` and `(-b)/(a)` are the roots of the given equation.


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