Question

If $x$ is real, then the expression
$\frac {x^2 +34x-71}{x^2+2x-7}$

• A. lies between $4$ and $7$
• B. lies between $5$ and $9$
• C. has no value between $4$ and $7$
• D. has no value between $5$ and $9$

Answer 1

Answer: (D)

Let $\frac {x^2 +34x-71}{x^2+2x-7} = m$
$\therefore \, x^2 + 34x - 71 = mx^2 + 2mx - 7m$
$\Rightarrow \, x^2 (1 - m) + (34 - 2m)x + 7m - 71 = 0 $
Since $x$ is real
$\therefore $ Disc $\geq$ 0
$\therefore \, (34 - 2m)^2 - 4(1 - m) (7m - 71) \geq 0$
$ (17 - m)^2 - (1 - m) (7m - 71) \geq 0 $
$\Rightarrow \, 289 + m^2 - 34m - [7m - 71 - 7m^2 + 71m] \geq 0$
$\Rightarrow \, 8m^2 - 112m + 360 \geq 0$
$ \Rightarrow \,m^2 - 14m + 45 \geq 0$
$\Rightarrow \, (m - 9) (m - 5) \geq 0$
$\Rightarrow \, m > 9, m > 5 $ or $m < 9, m < 5$.
$\Rightarrow \, m > 9$ or $m < 5$
$\therefore \, m$ has no value between $5$ and $9$.