Question

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

15x² – 28 = x

Answer 1

Given: 15x² – 28 = x

Step-wise calculation:

1. Put in standard form:

15x² – x – 28 = 0

So a = 15, b = –1, c = –28.

2. Compute the discriminant:

D = b² – 4ac 

= (–1)² – 4(15)(–28) 

= 1 + 1680

= 1681

3. Note 1681 = 41², so `sqrt(D) = 41`.

4. By the discriminant test, D > 0 ⇒ Two distinct real roots; moreover if `sqrt(D)` is an integer while a, b, c are integers, then the expressions `(-b ± sqrt(D))/(2a)` are rational numbers. 

5. Applying that to this case:

Roots = `(-b ± sqrt(D))/(2a)`

= `(1 ± 41)/30`

= `42/30`

= `7/5`

And `(-40)/30 = (-4)/3`, both rational and not equal.

The roots are rational and unequal because the discriminant is positive so two distinct real roots and is a perfect square so the square root is an integer, making the quadratic-formula values rational.