Quadratic Equation and Inequalities Part 1

Given, p and q are positive numbers.
p+q=2 . . . (i)
p4+q4=272
⇒‌‌(p2+q2)2−2p2q2=272
⇒‌‌[(p+q)2−2pq]2−2p2q2=272
⇒‌‌[(2)2−2pq]2−2p2q2=272 [from Eq. (i)]
⇒‌‌(4−2pq)2−2p2q2=272
⇒16+4p2q2−16pq−2p2q2=272
⇒‌‌2p2q2−16pq−256=0
⇒‌‌p2q2−8pq−128=0
pq=‌

8±√64+4×128

2×1

=‌

8±√576

2

=‌

8±24

2

∴‌‌pq=16,−8
Here, pq=−8 is not possible as p and q are positive.
∴‌‌pq=16
Now, the equation whose roots are p and q is
x2−(p+q)x+pq=0
x2−2x+16=0