The least positive value of a for which the equation, 2x2+(a−10)x+332=2a has real roots is ____

2x2+(a−10x)+33/2=2a has real roots ⇒ Discriminant ≥0. ⇒(a−10)2−(4)(2)(33/2−2a)≥0⇒a2−20a+100−132+16a≥0 ⇒a2−4a−32≥0(a+4)(a−8)≥0⇒a≤−4 or a≥8∴ The leas positive value of a is 8.

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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 ____

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answer is 8.

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Detailed Solution

$2 x^{2} + (a - 10 x) + 33 / 2 = 2 a$ has real roots $\Rightarrow$ Discriminant $\geq 0$ .
$\Rightarrow {(a - 10)}^{2} - (4) (2) (33 / 2 - 2 a) \geq 0 \Rightarrow a^{2} - 20 a + 100 - 132 + 16 a \geq 0$
$\Rightarrow a^{2} - 4 a - 32 \geq 0 (a + 4) (a - 8) \geq 0 \Rightarrow a \leq - 4 o r a \geq 8$
$∴$ The leas positive value of a is 8.