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Find the values of k for which the equations x2 – kx – 21 = 0 and x2 – 3kx + 35 = 0 will have a common root?

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Let α be the common root of both the given equations. Then a satisfies both the equations. So, 

α2 – kα – 21 = 0                              ...(i) 

a2 – 3kα + 35 = 0                          ...(ii) 

Solving equations (i) and (ii) simultaneously, we get

\(\frac{α^2}{-35k-63k}=\frac{α}{-21-35}=\frac{1}{-3k+k}\)

⇒ α2\(\frac{-98k}{-2k}\) = 49 and α = \(\frac{-56k}{-2k}\) = \(\frac{28}{k}\)

∴ 49 = \(\big(\frac{28}{k}\big)^2\) ⇒ k2\(\frac{28\times28}{49}\) = 16 ⇒ k = ± 4.

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