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प्रश्न
If the roots of the equations `ax^2+2bx+c=0` and `bx^2-2sqrtacx+b=0`are simultaneously real then prove that `b^2=ac`
उत्तर
It is given that the roots of the equation `ax^2+2bx+c=0` are real
∴` D_1=(2b)^-4xxaxxc=0` are real.
⇒`4(b^2-ac)≥0`
⇒` -4(b^2-ac)≥0`
⇒ b^2-ac≥0 ................(1)
Also, the roots of the equation `bx^2-2sqrtacx+b=0` are real.
∴` D_2=(-2sqrtac)^2-4xxbxxb≥0`
⇒`4(ac-b^2)≥0`
⇒`-4(b^2-ac)≥0`
⇒`b^2-ac≥0` ....................(2)
The roots of the given equations are simultaneously real if (1) and (2) holds true together.
This is possible if
`b^2-ac=0`
⇒`b^2=ac`
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