Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.2 AIME, harder Prove it

Example 2. The ratio of the two roots of the equation $a x^{2}+b x+c=0$ is $2: 3$ , prove that $6 b^{2}=25 a_{c}$ .

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

(Note: The notation $a_{c}$ in the original problem seems to be a typo, it should be $ac$ . The correct statement should be $6b^2 = 25ac$ .)

Solution

Solution: Let the two roots be $x_{1}, x_{2}$ , then $x_{1}=2 x_{2} / 3$ . We have
$\frac{2}{3} b^{2}-\left(\begin{array}{l} 2 \\ 3 \end{array}+1\right)^{2} a c=0,$

which is $6 b^{2}=25 a c$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.