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Nature of Roots

Find the numb...

Question

Find the number of real roots of the equation
$Δ (x) = ∣ ∣ ∣ ∣ \begin{matrix} 6 - 5 x & x - 2 & 4 - 4 x x - 2 & 2 - x & 0 4 - 4 x & 0 & 3 - 5 x \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Solution

$Δ (x) = ∣ ∣ ∣ ∣ \begin{matrix} 6 - 5 x & x - 2 & 4 - 4 x x - 2 & 2 - x & 0 4 - 4 x & 0 & 3 - 5 x \end{matrix} ∣ ∣ ∣ ∣ = 0 u s i n g C_{1} \to C_{1} + C_{2} - C_{3} w e g e t Δ (x) = ∣ ∣ ∣ ∣ \begin{matrix} 0 & 0 & 4 - 4 x 0 & 2 - x & 0 1 + x & 0 & 3 - 5 x \end{matrix} ∣ ∣ ∣ ∣ = (1 + x) (2 - x) (4 x - 4) = 0 ∴ Δ (x) = 0 h a s t h r e e r e a l r o o t s$

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