Question

If the root of the equation ax²+bx+c = 0 (a ≠0) be in the ratio of 3:4, shoe that: 12b² = 49ac. ​

Answer 1

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Let the roots of the equation ax²+bx+c = 0 be 3k and 4k (where k is some constant).

By Vieta's formulas, we know that the sum of the roots of a quadratic equation is equal to -b/a and the product of the roots is equal to c/a.

Thus, we have:

3k + 4k = -b/a

7k = -b/a

k = -b/7a

3k * 4k = c/a

12k² = c/a

12(-b/7a)² = c/a

12b²/49a² = c/a

Multiplying both sides by 49a², we get:

12b² = 49ac

Therefore, we have shown that if the roots of the quadratic equation ax²+bx+c = 0 are in the ratio of 3:4, then 12b² = 49ac.

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