The least positive value of 'a' for which the equation, 2x^{2}+(a-10)x+\frac{33}{2}=2a has real roots is
Option: 1 8
Option: 2 6
Option: 3 4
Option: 42
 

Answers (1)

 

 

Nature of Roots -

Let the quadratic equation is ax2 + bx + c = 0

D is the discriminant of the equation.

 

ii) If D > 0, then roots will be real and distinct. 

\\\mathrm{x_1 = \frac{-b + \sqrt{D}}{2a} } \;\mathrm{and \;\;x_2 = \frac{-b - \sqrt{D}}{2a} } \\\\\mathrm{Then,\;\; ax^2+bx +c =a(x-x_1)(x-x_2) }

 

iii) if roots D = 0, then roots will be real and equal, then


\\\mathrm{x_1=x_2 = \frac{-b}{2a} } \\\mathrm{Then, \;\; ax^2+bx +c =a(x-x_1)^2 }

-

{D \geqslant 0} \\\\ {(a-10)^{2}-8\left(\frac{33}{2}-2 a\right) \geq 0} \\\\ {a^{2}+100-20 a-132+16 a \geq 0}

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

a \leq -4 \ \text{ or }\ a \geq \ 8

least positive value is 8.

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