Question

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

16x² = 24x + 1

Answer 1

Given: 16x² = 24x + 1

Step-wise calculation:

1. Put in standard form:

16x² – 24x – 1 = 0

So a = 16, b = –24, c = –1.

2. Discriminant: Δ = b² – 4ac

= (–24)² – 4(16)(–1) 

= 576 + 64

= 640

Since Δ > 0, the equation has two distinct real roots.

3. Simplify `sqrt(Δ)`:

`sqrt(640) = sqrt(64 xx 10)`

 = `8sqrt(10)`

Because 10 is not a perfect square, `sqrt(10)` is irrational, so `sqrt(Δ)` is irrational.

4. By the quadratic formula the roots are `(-b ± sqrt(Δ))/(2a)`. 

Here `(-b)/(2a) = 24/32 = 3/4` is rational, while `sqrt(Δ)/(2a) = (8sqrt(10))/(32) = (sqrt(10))/4` is irrational.

Adding or subtracting a nonzero irrational to a rational gives two irrational numbers; the ± gives two different values.

Therefore, the two roots are irrational and unequal.