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

imaginary roots
B

real roots
C

purely 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,
$D_{1} + D_{2} = b_{1}^{2} - 4 c_{1} + b_{2}^{2} - 4 c_{2}$
$\begin{array}{l} = (b_{1}^{2} + b_{2}^{2}) - 4 (c_{1} + c_{2}) \\ = b_{1}^{2} + b_{2}^{2} - 2 b_{1} b_{2} [∵ b_{1} b_{2} = 2 (c_{1} + c_{2})] \\ = {(b_{1} - b_{2})}^{2} \geq 0 \end{array}$
$\Rightarrow D_{1} \geq 0 or D_{2} \geq 0 or D_{1} and D_{2}$ both are positive
Hence, at least one of the equations has real roots.

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