1.

Solve the x by quadratic formula `p^2x^2+(p^2-q^2)x-q^2=0`

Answer» The given equation is `p^(2)x^(2)+(p^(2)-q^(2))x-q^(2)=0.`
Comparing in with `ax^(2)+bx+c=0,` we get
`a=p^(2),b=(p^(2)-q^(2))" and "c=-q^(2).`
`:." "D=(b^(2)-4ac)=(p^(2)-q^(2))^(2)-4xxp^(2)xx(-q^(2))`
`=(p^(2)-q^(2))^(2)+4p^(2)q^(2)=(p^(2)+q^(2))^(2)gt0.`
So, the given equation has real roots.
Now, `sqrt(D)=(p^(2)+q^(2)).`
`:." "alpha=(-b+sqrt(D))/(2a)=(-(p^(2)-q^(2))+(p^(2)+q^(2)))/(2p^(2))=(2p^(2))/(2p^(2))=(q^(2))/(p^(2)),`
`beta=(-b-sqrt(D))/(2a)=(-(p^(2)-q^(2))-(p^(2)-q^(2)))/(2p^(2))=(-2p^(2))/(2p^(2))=-1.`
Hence, `(q^(2))/(p^(2))` and -1 are the roots of the given equation.


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