If f(x) = x^{2} – 3x + 4 and f(x) = f(2x + 1), find the values of x.

Given: f(x) = x^{2} – 3x + 4 ----- (1)

and f(x) = f(2x + 1)

Need to Find: Value of x

Replacing x by (2x + 1) in equation (1) we get,

f(2x + 1) = (2x + 1)^{2} – 3(2x + 1) + 4 ----- (2)

According to the given problem, f(x) = f(2x + 1)

Comparing (1) and (2) we get,

x^{2} – 3x + 4 = (2x + 1)^{2} – 3(2x + 1) + 4

⇒ x^{2} – 3x + 4 = 4x^{2} + 4x + 1 – 6x – 3 + 4

⇒ 4x^{2} + 4x + 1 – 6x – 3 + 4 – x^{2} + 3x – 4 = 0

⇒ 3x^{2} + x – 2 = 0

⇒ 3x^{2} + 3x – 2x – 2 = 0

⇒ 3x(x + 1) – 2(x + 1) = 0

⇒ (3x – 2)(x + 1) = 0

So, either (3x – 2) = 0 or (x + 1) = 0

Therefore, the value of x is either or -1 [Answer]

1