The value of c for which |α2−β2|=74 where α and β are the roots of 2x2+7x+c=0 is

α and β are roots of 2x2+7x+c=0α+β=−72, αβ=c2(α−β)2=(α+β)2−4αβ=494−2c∴ |α2−β2|=74⇒ |(α−β)(α+β)|=74⇒ | 494−2c×−72|=74⇒ |49−8c×74|=74⇒ |49−8c|=1⇒ 49−8c=1⇒8c=48⇒c= 484=6

The value of c for which |α2−β2|=74 where α and β are the roots of 2x2+7x+c=0 is

α and β are roots of 2x2+7x+c=0α+β=−72, αβ=c2(α−β)2=(α+β)2−4αβ=494−2c∴ |α2−β2|=74⇒ |(α−β)(α+β)|=74⇒ | 494−2c×−72|=74⇒ |49−8c×74|=74⇒ |49−8c|=1⇒ 49−8c=1⇒8c=48⇒c= 484=6

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$T h e v a l u e o f c f o r w h i c h | α^{2} - β^{2} | = \frac{7}{4} w h e r e α a n d β a r e t h e r o o t s o f 2 x^{2} + 7 x + c = 0 i s$

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Detailed Solution

$\begin{array}{l} α a n d β a r e r o o t s o f 2 x^{2} + 7 x + c = 0 \\ α + β = \frac{- 7}{2}, α β = \frac{c}{2} \\ {(α - β)}^{2} = {(α + β)}^{2} - 4 α β \\ = \frac{49}{4} - 2 c \\ ∴ | α^{2} - β^{2} | = \frac{7}{4} \\ \Rightarrow | (α - β) (α + β) | = \frac{7}{4} \\ \Rightarrow | \sqrt{\frac{49}{4} - 2 c} \times \frac{- 7}{2} | = \frac{7}{4} \\ \Rightarrow | \sqrt{49 - 8 c} \times \frac{7}{4} | = \frac{7}{4} \\ \Rightarrow | \sqrt{49 - 8 c} | = 1 \\ \Rightarrow 49 - 8 c = 1 \\ \Rightarrow 8 c = 48 \\ \Rightarrow c = \frac{48}{4} = 6 \end{array}$