Question

The coefficients $a,b,c$ of the quadratic equation $a{x}^2+bx+c=0$ are determined throwing a die three times.Find the probabilities that the roots are real

• A. $\frac{42}{216}$
• B. $\frac{28}{216}$
• C. $\frac{38}{216}$
• D. $\frac{32}{216}$

Answer 1

The correct option is A $\frac{42}{216}$

for real solution
$b^2>4ac$

b	(a,c)	total
1	--	0
2	(1,1)	1
3	(1,1)(1,2)(2,1)	3
4	(1,1)(1,2)(2,1)(1,3)(3,1)(1,4)(4,1)	7
5	(1,1)(1,2)(2,1)(1,3)(3,1)(2,2)(2,3)(3,2)(1,4)(4,1)(1,5)(5,1)(1,6)(6,1)	14
6	(1,1)(1,2)(2,1)(1,3)(3,1)(2,2)(2,3)(3,2)(1,4)(4,1)(1,5)(5,1)(1,6)(6,1)(2,4)(4,2)(3,3)	17
	Total	42

Total no. of ways of choosing a,b,c = $6^3=216$
$P=\frac{42}{216}$