Question

If f(x) = x² – 3x + 4, then find the values of x satisfying the equation f(x) = f(2x + 1).

Answer 1

Given,

f(x) = x² – 3x + 4.

We need to find x satisfying 

f(x) = f(2x + 1). 

We have,

f(2x + 1) = (2x + 1)² – 3(2x + 1) + 4 

⇒ f(2x + 1) = (2x)² + 2(2x)(1) + 1² – 6x – 3 + 4 

⇒ f(2x + 1) = 4x² + 4x + 1 – 6x + 1 

∴ f(2x + 1) = 4x² – 2x + 2 

Now, 

f(x) = f(2x + 1) 

⇒ x² – 3x + 4 = 4x² – 2x + 2 

⇒ 3x² + x – 2 = 0 

⇒ 3x² + 3x – 2x – 2 = 0 

⇒ 3x(x + 1) – 2(x + 1) = 0 

⇒ (x + 1)(3x – 2) = 0 

⇒ x + 1 = 0 or 3x – 2 = 0 

⇒ x = –1 or 3x = 2 

∴ x = –1 or \(\frac{2}{3}\)

Thus,

The required values of x are –1 and \(\frac{2}{3}\).