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. Therefore, we may factor the given expression.

Expanding $z(z+i)(z+3i)=2002i$ , we have $z^3 + 4iz^2 - 3z - 2002i = 0$ . We may factor it as $(z^3 + 14z^2) - 10i(z^2 + 14iz) - 143(z + 14i) = (z + 14i)(z^2 - 10iz -143) = 0$

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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