Two numbers b and c are chosen at random (with$$ replacement from the numbers $$1$$,$$2$$,$$3$$,$$4$$,$$5$$,$$6$$,$$7$$,$$8$$ and $$9). The probability that $$x2+bx+c$$$$0for$$ all x∈R is

Question

Two numbers b and c are chosen at random (with$$ replacement from the numbers $$1$$,$$2$$,$$3$$,$$4$$,$$5$$,$$6$$,$$7$$,$$8$$ and $$9). The probability that $$x2+bx+c$$>$$0for$$ all x∈R is

Infinity Learn · Accepted Answer

Here,$$x2+bx+c$$>$$0$$∀x∈R∴ D<$$0$$⇒ $$b2$$<$$4cvalue$$ of b possible values of $$c1$$ $$1$$<$$4c$$ ⇒c>$$14$$ ⇒$${1$$,$$2$$,$$3$$,$$4$$,$$5$$,$$6$$,$$7$$,$$8$$,$$9}2$$ $$4$$<$$4c$$ ⇒c>$$1$$ ⇒$${2$$,$$3$$,$$4$$,$$5$$,$$6$$,$$7$$,$$8$$,$$9}3$$ $$9$$<$$4c$$ ⇒c>$$94$$ ⇒$${3$$,$$4$$,$$5$$,$$6$$,$$7$$,$$8$$,$$9}4$$ $$16$$<$$4c$$ ⇒c>$$4$$ ⇒$${5$$,$$6$$,$$7$$,$$8$$,$$9}5$$ $$25$$<$$4c$$ ⇒c>$$625$$ ⇒$${7$$,$$8$$,$$9}6$$ $$36$$<$$4c$$ ⇒c>$$9$$ Impossible $$7$$ impossible $$8$$ impossible $$9$$ impossibleNumber of favorable $$ca8e8$$ $$=9$$ $$+$$ $$8$$ $$+$$ $$7+6+3=32Total$$ $$ways=9x9=81Required$$ probability $$=$$ $$32/81$$

Two numbers b and c are chosen at random (with replacement from the numbers 1,2,3,4,5,6,7,8 and 9). The probability that $x^{2} + bx + c > 0$ for all $x \in R$ is

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Two numbers b and c are chosen at random (with replacement from the numbers 1,2,3,4,5,6,7,8 and 9). The probability that x2+bx+c>0for all x∈R is

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Two numbers b and c are chosen at random (with replacement from the numbers 1,2,3,4,5,6,7,8 and 9). The probability that $x^{2} + bx + c > 0$ for all $x \in R$ is