First slide

Theory of equations

Question

If the quadratic equations, $x^{2} + bx + c = 0 and {bx}^{2} + cx + 1 = 0$ have a common root then

Moderate

A

b + c + 1 = 0
B

$b^{2} + c^{2} - 1 = bc - b - c$
C

b + c - 1 = 0
D

$b^{2} + c^{2} + 1 = bc + b + c$

Solution

Equations $x^{2} + bx + c = 0 and {bx}^{2} + cx + 1 = 0$ have a common root.
$\begin{array}{l} (bc - 1)^{2} = (c - b^{2}) (b - c^{2}) \\ \Rightarrow b^{2} c^{2} + 1 - 2 bc = bc - b^{3} - c^{3} + b^{2} c^{2} \\ \Rightarrow 1 + c^{3} + b^{3} = 3 bc \\ \Rightarrow (1 + c + b) (c^{2} + b^{2} + 1 - b - c - bc) = 0 \\ \Rightarrow b + c + 1 = 0 and c^{2} + b^{2} + 1 = c + b + bc \end{array}$

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