HMMT February

Algebra Difficulty 4.5 AIME Find the answer

Find all possible values of $\frac{d}{a}$ where $a^{2}-6 a d+8 d^{2}=0, a \neq 0$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Dividing $a^{2}-6 a d+8 d^{2}=0$ by $a^{2}$ , we get $1-6 \frac{d}{a}+8\left(\frac{d}{a}\right)^{2}=0$ . The roots of this quadratic are $\frac{1}{2}, \frac{1}{4}$ .

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