Question

If the roots of equation $x^2+a^2=8x+6a$ are real, then

• A. $a\in [2,8]$
• B. $a\in [-2,8]$
• C. $a\in (2,8)$
• D. $a\in (-2,8)$

Answer 1

Answer: (B)

$a\in [-2,8]$

Given equation,

$x^2-8x+a^2-6a=0$,

We know that,

$D=b^2-4ac$

here, $a=1,b=-8,c=a^2-a$

For real roots,

$D\ge 0$

$\Rightarrow 64-4(a^2-6a)\ge 0$

$\Rightarrow 4a^2+24a-64\le 0$

$\Rightarrow (a+2)(a-8)\le 0$

$\Rightarrow a\in [-2,8]$.

Hence, option ${}^{\prime }{B}^{\prime }$ is correct.