1. Tardigrade
  2. Question
  3. Mathematics
  4. If one root of the quadratic equation px 2+ qx + r =0( p ≠ 0) is a surd of the form (√ a /√ a +√ a - b ) where p , q , r ∈ Q and a , b are positive integers which are not perfect square of integer, then the other root is

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Q. If one root of the quadratic equation $px ^2+ qx + r =0( p \neq 0)$ is a surd of the form $\frac{\sqrt{ a }}{\sqrt{ a }+\sqrt{ a - b }}$ where $p , q , r \in Q$ and $a , b$ are positive integers which are not perfect square of integer, then the other root is

Complex Numbers and Quadratic Equations

Solution:

$\alpha=\frac{\sqrt{a}}{\sqrt{a}+\sqrt{a-b}}=\frac{\sqrt{a}(\sqrt{a}-\sqrt{a-b})}{a-(a-b)}=\frac{a-\sqrt{a(a-b)}}{b}$
Conjugate of $\alpha$ is $\frac{a+\sqrt{a(a-b)}}{b} \Rightarrow(C)$
$=\frac{a^2-(a(a-b))}{b(a-\sqrt{a(a-b)})}=\frac{a}{a-\sqrt{a(a-b)}}=\frac{\sqrt{a}}{\sqrt{a}-\sqrt{a-b}} \Rightarrow (A)$