The quadratic equations $$x2$$−$$6x$$ $$+$$ a $$=$$ $$0$$ and $$x2$$−cx $$+$$ $$6$$ $$=$$ $$0$$ have one root in common. The other roots of the first and second equations are integers in the ratio $$4$$:$$3$$. Then the common root is

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The  quadratic equations $$x2$$−$$6x$$ $$+$$ a $$=$$ $$0$$      and $$x2$$−cx $$+$$ $$6$$ $$=$$ $$0$$       have one  root in common. The other roots of the first and second equations are  integers in the ratio $$4$$:$$3$$. Then the common root is

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The given quadratic equations  are $$x2$$−$$6x$$ $$+$$ a $$=$$ $$0$$     …… $$($$ $$1$$ $$)$$    and $$x2$$−cx $$+$$ $$6$$ $$=$$ $$0$$       ….. $$($$ $$2$$ $$)Given$$ that one root  is common, it is consider as α     And the other roots of the first and  second equations are integers in the ratio $$4$$ : $$3$$∴      the roots are α,$$4$$β,α,$$3$$β       From Eq.$$($$ $$1$$ $$)$$ &  Eq.$$($$ $$2$$ $$)$$ consider product of the roots ⇒$$4$$αβ$$=a$$    and  $$3$$αβ$$=6$$  ∴αβ$$=2$$, $$a=8$$     From $$($$ $$1$$ $$)$$: $$x2$$−$$6x$$ $$+$$ $$8$$ $$=$$ $$0$$     ⇒$$x2$$−$$4x$$−$$2x+8=0$$     ⇒(x$$−$$2)(x$$−$$4)=0$$     ∴$$x=2$$,$$4$$         ….. $$($$ $$3$$ $$)If$$  take $$2$$ is common root then  Eq. $$($$ $$2$$ $$) will be $$x2$$−cx $$+$$ $$6$$ $$=$$ $$0$$       ⇒(2)2$$−$$c(2)+6=0$$     ⇒$$c=5$$     .        If  take $$4$$ is common root then Eq.$$($$ $$2$$ $$)is$$ not  satisfies.∴     Eq. $$($$ $$2$$ $$) ⇒     $$x2$$−$$5x+6=0$$     ⇒$$x2$$−$$3x$$−$$2x+6=0$$     ⇒(x$$−$$3)(x$$−$$2)=0$$     ∴$$x=3$$,$$2$$         ….. $$($$ $$4$$ $$)From$$ Eq.$$($$ $$3$$ $$) &  Eq.$$($$ $$4$$ $$)$$∴     common root $$=2$$

The quadratic equations x2−6x + a = 0 and x2−cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. Then the common root is

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