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

43 / 216

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174 / 216

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178 / 216

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48 / 216

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Solution

The correct option is B $174 / 216$
P(imaginary roots) $= 1 -$ P(real roots)
for real solution
$b^{2} > 4 a c$
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 $a, b, c$ = $6^{3} = 216$
$P (r e a l r o o t s) = \frac{42}{216}$
$P (i m r o o t s) = \frac{174}{216}$

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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

The coefficients a,b,c of the quadratic equation ax2+bx+c=0 are determined by throwing a die three times. Find the probabilities such that roots are imaginary.

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