Question

Discuss the nature of the roots of the following equation without actually solving it: 

25x² + 30x + 7 = 0

Answer 1

Given: 25x² + 30x + 7 = 0

Step-wise calculation:

1. Identify coefficients: a = 25, b = 30, c = 7.

2. Compute discriminant: Δ = b² – 4ac

= 30² – 4 × 25 × 7

= 900 – 700

= 200

3. Discriminant test: Because Δ > 0 the roots are real and unequal (distinct).

4. Check whether the roots are rational: The quadratic formula uses `±sqrt(Δ)`. 

Here, `sqrt(Δ) = sqrt(200) = 10sqrt(2)`, which is not an integer rational number. 200 is not a perfect square, so the `±sqrt(Δ)` part is irrational and therefore each root is irrational.

Because Δ = 200 > 0 the equation has two distinct real roots (unequal) and because `sqrt(200)` is not a perfect square the two roots are irrational. 

Hence, irrational and unequal is correct.