Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

16x² = 24x + 1

Answer 1

Given: 16x² = 24x + 1

Step-wise calculation:

1. Bring to standard form:

16x² – 24x – 1 = 0

So a = 16, b = –24, c = –1.

2. Use the quadratic formula `x = (-b ± sqrt(b^2 - 4ac))/(2a)`.

3. Compute the discriminant:

D = (–24)² – 4(16)(–1) 

= 576 + 64

= 640

4. `sqrt(D) = sqrt(640)`

= `sqrt(64 xx 10)` 

= `8sqrt(10)`

5. `x = (24 ± 8sqrt(10))/(32)` 

= Divide numerator and denominator by 8 

= `(3 ± sqrt(10))/4`

The two real roots are `x = ((3 + sqrt(10)))/4` and `x = ((3 - sqrt(10)))/4`.