Question

If p, q, r are real numbers satisfying the condition p + q + r = 0, then the roots of the quadratic equation 3px² + 5qx + 7r = 0 are

1) Positive
2) Negative
3)Real and Distinct
4)Imaginary​

Answer 1

Given : p, q , r are real numbers satisfying the condition p + q + r = 0,

To check : the roots of quadratic equation 3px² + 5qx + 7r = 0 are

1) Positive

2) Negative

3)Real and Distinct

4)Imaginary

solution : here, p + q + r = 0

⇒r = -(p + q) .....(1)

now quadratic equation, 3px² + 5qx + 7r = 0

⇒3px² + 5qx - 7(p + q) = 0 [ from eq (1). ]

Discriminant = (5q)² - 4 × 3p × {-7(p + q)}

= (5q)² + 84p(p + q)

= (5q)² + 84p² + 84pq

= (5q)² + (42/5 p)² + 2 × 5q × 42/5 p - (42/5 p)² + 84p²

= (5q + 42/5 p)² + {84 - (42/5)²}p² > 0

Therefore the roots of given quadratic equation are real and distinct. i.e., option (3) is correct choice.

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