Question

If $a$ be chosen at random in the interval $[0,5]$, show that the probability that the equation $4x^2+4ax+(a+2)=0$ to have real roots is $\frac{3}{5}$

Answer 1

Equation $4x^2+4ax+(a+2)=0$ to have real roots,if its discriminant $D\ge 0$

$\Rightarrow 16a^2-16(a+2)\ge 0$

$\Rightarrow a^2-a-2\ge 0$

$\Rightarrow a^2-2a+a-2\ge 0$

$\Rightarrow a(a-2)+1(a-2)\ge 0$

$\Rightarrow (a-2)(a+1)\ge 0$

$\therefore a>2ora<-1$

But in the interval $[0,5],a>2\Rightarrow a\in [2,5]$

$\therefore$probability$=\frac{lengthofinterval[2,5]}{lengthofinterval[0,5]}=\frac{5-2}{5-0}=\frac{3}{5}$