Question

If 'x' be real, prove that,

$\frac{ {x}^{2} + 34x - 71 }{ {x}^{2} + 2x - 7 }$

can have no value between 5 and 9.


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Answer 1

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Answer 2

Step-by-step explanation:

Given: (x² + 34x - 71)/(x² + 2x - 7)

Let y = (x² + 34x - 71)/(x² + 2x - 7)

⇒ x²y + 2xy - 7y = x² + 34x - 71

⇒ (y - 1)x² + 2(y - 17)x - (7y - 71) = 0

Given that x is real. x ∈ R.

Δ ≥ 0

⇒ 4(y - 17)² + 4(y - 1)(7y - 71) ≥ 0

⇒ y² - 34y + 289 + 7y²- 7y - 71y + 71 ≥ 0

⇒ 8y² - 112y + 360 ≥ 0

⇒ y² - 14y + 45 ≥ 0

⇒ y² - 9y - 5y + 45 ≥ 0

⇒ y(y - 9) - 5(y - 9) ≥ 0

⇒ (y - 5)(y - 9) ≥ 0

⇒ y ∈ (-∞,5] ∪ [9,∞)

⇒ y ∉ (5,9)

∴ Expression does not lie between 5 and 9.

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