1984 AHSME Problems/Problem 24

Problem

If $a$ and $b$ are positive real numbers and each of the equations $x^2+ax+2b=0$ and $x^2+2bx+a=0$ has real roots, then the smallest possible value of $a+b$ is

$\mathrm{(A) \ }2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ } 6$

Solution

Since both of the equations have real roots, both of their discriminants are nonnegative. Therefore, we have

$a^2-4(1)(2b)=a^2-8b\geq0\implies a^2\geq8b$ from the first equation, and

$(2b)^2-4(1)(a)=4b^2-4a\geq0\implies 4b^2\geq4a\implies b^2\geq a$ from the second.

We can square the second equation to get $b^4\geq a^2$ , and combining this with the first one gives $b^4\geq a^2\geq8b$ , so $b^4\geq8b$ . We can divide both sides by $b$ , since it is positive, and take the cubed root of that to get $b\geq2$ . Therefore, we have $a^2\geq8b\geq8(2)=16$ , and since $a$ is positive, we can take the square root of this to get $a\geq4$ . Therefore, $a+b\geq 2+4=6$ , and the smallest possible value is $6, \boxed{\text{E}}$ .

1984 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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1984 AHSME Problems/Problem 24

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