Question

if p and q are the roots of the equation x^2 +px - q = 0 then find the values of p and q

Answer 1

x^2+px-q=0

Considering p as a root

${p}^{2} + p(p) - q = 0 \\ \\ \\ {p}^{2} + {p}^{2} - q = 0 \\ \\ \\ 2 {p}^{2} - q = 0 \\ \\ \\ q = 2 {p}^{2} ..........................(1)$

Now considering q as a root

${q}^{2} + pq - q = 0 \\ \\ \\ q(q + p - 1) = 0 \\ \\ \\ p + q = 1...................(2)$
Now from (1)

$p + {2p}^{2} = 1 \\ \\ {2p}^{2} + p - 1 = 0 \\ \\ \\ {2p}^{2} + 2p - p - 1 = 0 \\ \\ \\ 2p(p + 1) - 1(p + 1) = 0 \\ \\ \\ (2p - 1) (p + 1) = 0 \\ \\ \\ 2p - 1 = 0 \: \: \: \: \: \: or \: \: \: \: \: p + 1 = 0 \\ \\ \\ p = \frac{1}{2} \: \: \: \: \: \: or \: \: \: \: \: \: p = - 1$
Now from (2)

p+q=1

Considering p as 1/2

$\frac{1}{2} + q = 1 \\ \\ q = 1 - \frac{1}{2} \\ \\ q = \frac{1}{2}$
Considering p as -1


p+q = 1

-1 + q = 1

q = 1+1

q = 2


ANS:- The values of p and q are 1/2 or -1 and 1/2 or 2

Answer 2

Answer:

Step-by-step explanation:

x^2 + px - q = 0

put x = q

(q)^2 + p(q) - q = 0

q^2 + pq - q = 0

q(q + p - 1) = 0

q + p = 1

p = 1 - q   ....(1)

now put x = p

(p)^2 + p(p) - q = o

p^2 + p^2 - q = 0

2p^2 - q = 0

q = 2p^2 .....(2)

put the equation 2 in equation 1

p = 1 - 2p^2

2p^2 + p - 1 = 0

2p^2 + (2p - p) - 1 = 0

2p^2 + 2p - p - 1 = 0

2p(p + 1) - 1(p+1) = 0

(2p - 1)     (p + 1)

2p - 1 = 0        p + 1 = 0

2p = 1              p = -1

p = 1/2