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JEE MainMathematicsToughQuadratic Equations

The values of for which roots of the equation
(a−2)x2−(a−4)x−2=0(a - 2)x^{2} - (a - 4)x - 2 = 0 are real and distinct and their
difference is greater than or equal to 3, are

Show answer and solution
D>0⇒a2>0⇒a∈R−{0,2}D > 0 \Rightarrow a^{2} > 0 \Rightarrow a \in R - \{ 0,2\}
∣A−β∣≥3|Α - \beta| \geq 3
⇒(α−−4αβ≥9\Rightarrow (\alpha - - 4\alpha\beta \geq 9
⇒(a−4a−2)2+4.2a−2≥9\Rightarrow \left( \frac{a - 4}{a - 2} \right)^{2} + \frac{4.2}{a - 2} \geq 9
(2a−3)(a−3)≤0(2a - 3)(a - 3) \leq 0
⇒a∈[]\Rightarrow a \in \left\lbrack \right\rbrack

Source: Grass-G / M-19 Mathematics Paper 28 Nov Fourth.docx

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