First slide

Theory of equations

Question

If $b_{1} b_{2} = 2 (c_{1} + c_{2}),$ then at least one of the equations $x^{2} + b_{1} x + c_{1} = 0$ and $x^{2} + b_{2} x + c_{2} = 0$ has

Moderate

A

Real roots
B

Purely imaginary roots
C

Imaginary roots
D

None of these

Solution

Let $D_{1}$ and $D_{2}$ be discriminants of $x^{2} + b_{1} x + c_{1} = 0$
and $x^{2} + b_{2} x + c_{2} = 0$ respectively. Then
$D_{1} + D_{2} = {b_{1}}^{2} - 4 c_{1} + {b_{2}}^{2} - 4 c_{2} = ({b_{1}}^{2} + {b_{2}}^{2}) - 4 (c_{1} + c_{2}) {b_{1}}^{2} + {b_{2}}^{2} - 2 b_{1} b_{2}, (∵ b_{1} b_{2} = 2 (c_{1} + c_{2})) = {(b_{1} - b_{2})}^{2} \geq 0 \Rightarrow D_{1} \geq 0 or D_{2} \geq 0 or D_{1} and D_{2} both are positive .$

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