Question

If x2 - ax + b = 0 and x2 - px + q = 0 have a root in common and the second equation has equal roots, find the value
of b + q.​

Answer 1

Answer:

Given equations are: x

2

−ax+b=0 (i)

and x

2

−px+q=0 (ii)

Let α be the common root. Then roots of equation (ii) will be α and α. Let β be the other root of equation (i).

Thus roots of equation (i) are α,β and those of equation (ii) are α,α

Now α+β=a (iii)

αβ=b (iv)

2α=p (v)

α

2

=q (vi)

L.H.S.=b+q=αβ+α

2

=α(α+β) (vii)

and R.H.S.=

2

ap

=

2

(α+β)2α

=α(α+β) (viii)

from (vii) and (viii), L.H.S.=R.H.S.

Answer 2

Answer:

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Given equations are,

${x}^{2} - </strong><strong>a</strong><strong>x + </strong><strong>b</strong><strong> = 0.......(1)$

${x}^{2} - px + q = 0.......(2)$

Let $\alpha$be the common root. Then roots of equation (ii) will be $\alpha$and $\alpha$. Let $\beta$ be the other root of equation(i). Thus roots of equation (i) are $\alpha$,$\beta$ and those of equation (ii) are $\alpha$,$\alpha$

$now \: \alpha + \beta = a.....(3)$

$\alpha \beta = b......(4)$

$2 \alpha = p......(5)$

${ \alpha }^{2} = q......(6)$

$b + </em><em>q</em><em> = \alpha \beta + {a}^{2} \\ = a( \alpha + \beta )$

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