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
Therefore,
f (2x + 1) = (2x + 1)² – 3(2x + 1) + 4
= 4x² + 1 + 4x – 6x – 3 + 4
= 4x² – 2x + 2
Now,
f (x) = f (2x + 1)
⇒ x² – 3x + 4 = 4x² – 2x + 2
⇒ 4x² – x² – 2x + 3x + 2 – 4 = 0
⇒ 3x² + x – 2 = 0
⇒ 3x² + 3x – 2x – 2 = 0
⇒ 3x(x + 1) – 2(x +1) = 0
⇒ (3x – 2)(x +1) = 0
⇒ (x + 1) = 0 or ( 3x – 2) = 0
$\Rightarrow x=-1orx=\frac{2}{3}$
Hence, $x=-1,\frac{2}{3}$.