Resonating quadratics

Probability Level 4

Two numbers $b$ and $c$ are chosen at random (with the replacement from the numbers $1,2,3 \ldots 9$ ) . The probability that $x^{2} + bx + c > 0$ for all $x \in R$ .

$\text{Details and Assumptions :-}$ Let the probability could be represented as $\dfrac{A}{B}$ ,then find $A+B$ .

The answer is 113.

1 solution

Tran Quoc Dat
Apr 8, 2016

If $x^2+bx+c>0 \forall x$ , then $\Delta = b^2-4c<0$ or $b^2<4c \leq 4 \times 9 =36$ $\Rightarrow b<6$ .

If $b=1$ then $c \in \{1,2,\ldots,9\}$ . There are $9$ possible pairs.

If $b=2$ then $c \in \{2,3,\ldots,9\}$ . There are $8$ possible pairs.

If $b=3$ then $c \in \{3,4,\ldots, 9\}$ . There are $7$ possible pairs.

If $b=4$ then $c \in \{5,6,\ldots,9\}$ . There are $5$ possible pairs.

Lastly, if $b=5$ then $c \in \{7,8,9\}$ . There are $3$ possible pairs.

Number of possible pairs of $(b;c)$ equals $9+8+7+5+3=32$ . Overall there are $P^2_9+9=81$ (plus $9$ because there are $9$ pairs satisfy $b=c$ ).

The probability is $\frac{32}{81}$ , which means the answer is $\boxed{113}$ .

Resonating quadratics

The answer is 113.

1 solution

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