2002 AMC 12P Problems/Problem 23

Problem

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$ , where $a$ and $b$ are positive real numbers. Find $a.$

$\text{(A) }\sqrt{118} \qquad \text{(B) }\sqrt{210} \qquad \text{(C) }2 \sqrt{210} \qquad \text{(D) }\sqrt{2002} \qquad \text{(E) }100 \sqrt{2}$

Solution

Note that $2002 = 11 \cdot 13 \cdot 14$ . With this observation, it becomes easy to note that $z = -14i$ is a root of the given equation. However, it is not of the desired form in the problem, so we must factor the given expression to obtain the other 2 roots. From this point onwards, we assume that $z \neq -14i$ .

Expanding $z(z+i)(z+3i)=2002i$ , we have $z^3 + 4iz^2 - 3z - 2002i = 0$ . We may factor it as $(z + 14i)(z^2 - 10iz - 143) = 0$ . Since $z + 14i \neq 0$ , we must have $z^2 - 10iz - 143 = 0$ . Therefore, $z = \frac {10i \pm \sqrt{-100 + 4(143)}}{2} = 5i \pm \sqrt {118}$ .

Since $a > 0$ , we ignore the negative root. Therefore, $a = \boxed {\text{(A) } \sqrt {118}}$ .

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2002 AMC 12P Problems/Problem 23

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