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The roots of $a x^{2} + b x + c = 0$ are in the ratio 2 : 3. Which relation holds?

Competition practiceAlgebra

A $6 b^{2} = 25 a c$
B $25 b^{2} = 6 a c$
C $6 b^{2} = 5 a c$
D $b^{2} = 4 a c$

Step-by-Step Solution

Let the roots be

2 r

and

3 r

. Then

5 r = - b / a

and

6 r^{2} = c / a

. Substituting

r = - b / (5 a) : 6 b^{2} / (25 a^{2}) = c / a

, i.e.

6 b^{2} = 25 a c

.

Answer: A)

6 b^{2} = 25 a c

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