Question:
If every pair from among the equation $x^2 + px + qr = 0$, $x^2 + qx + rp = 0$ and $x^2 + rx + pq = 0$ has a common root, then the product of three common roots is.
If every pair from among the equation $x^2 + px + qr = 0$, $x^2 + qx + rp = 0$ and $x^2 + rx + pq = 0$ has a common root, then the product of three common roots is.
Updated On: Jul 6, 2022
- $pqr$
- $2pqr$
- $ p^2 q^2 r^2$
- none of these
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The Correct Option is A
Solution and Explanation
$\alpha \beta = qr, \beta \gamma = rp, \gamma \alpha = pq $
$\therefore \, (\alpha \beta ) (\beta \gamma ) (\gamma \alpha ) = (qr) (rp) (pq)$
$\Rightarrow \, (\alpha \beta \gamma)^2 = (pqr)^2$
$\Rightarrow \, \alpha \beta = pqr$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
