The least positive value of ' a ' for which the equation 2 x 2 + ( a - 10 ) x + 33 2 = 2 a has real roots is

Question

The least positive value of $' a'$ for which the equation $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is _______.

Smart Achievers · Accepted Answer

(8)

$Since, the given equation 2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ $has real roots.$

$∴$ $D \geq 0 \Rightarrow {(a - 10)}^{2} - 4 (2) (\frac{33}{2} - 2 a) \geq 0$

$\Rightarrow$ $a^{2} + 100 - 20 a - 132 + 16 a \geq 0$

$\Rightarrow$ $a^{2} - 4 a - 32 \geq 0 \Rightarrow (a - 8) (a + 4) \geq 0$

$\Rightarrow$ $a \in (- \infty, - 4] \cup [8, \infty)$

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