Question

If -4 is the root of the equation x2 + px – 4 = 0, and the equation x2 + px + q = 0 has equal roots, find the value of p and q

Answer 1

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

$let \alpha , \beta \: be \: roots \: of \: eq(i)$

ACCORDING TO QUESTION

$\alpha + \beta = { \alpha }^{2} + { \beta }^{2} .....(ii)$

$now \: term \: eq(i) \alpha + \beta = - p$

$\alpha \beta = q$

$from \: eq(ii)$

$- p = { \alpha }^{2} + { \beta }^{2} = ( \alpha + \beta {)}^{2} - 2 \alpha \beta$

$- p = ( - p {)}^{2} - 2q$

$- p = {p}^{2} - 2q$

${p}^{2} + p - 2q = 0$

Answer 2

Answer:

x

2

+px+q=0.....(i)

let \alpha , \beta \: be \: roots \: of \: eq(i)letα,βberootsofeq(i)

ACCORDING TO QUESTION

\alpha + \beta = { \alpha }^{2} + { \beta }^{2} .....(ii)α+β=α

2

+β

2

.....(ii)

now \: term \: eq(i) \alpha + \beta = - pnowtermeq(i)α+β=−p

\alpha \beta = qαβ=q

from \: eq(ii)fromeq(ii)

- p = { \alpha }^{2} + { \beta }^{2} = ( \alpha + \beta {)}^{2} - 2 \alpha \beta−p=α

2

+β

2

=(α+β)

2

−2αβ

- p = ( - p {)}^{2} - 2q−p=(−p)

2

−2q

- p = {p}^{2} - 2q−p=p

2

−2q

{p}^{2} + p - 2q = 0p

2

+p−2q=0