Evaluating rational algebraic expression

Question

If $x=1+\sqrt{3},$ find the value of $\dfrac{2x^2-4x+8}{3x^2-6x+10}.$

Answer 1

The equation is

$$\frac{2 \cdot (x-1)^2 + 6}{3 \cdot (x-1)^2 + 7} = 12/16 = 3/4$$

Answer 2

$(x-1)^2=3$ so $x^2= 2x+2$ so:

$$\dfrac{2x^2-4x+8}{3x^2-6x+10} =\dfrac{4x+4-4x+8}{6x+6-6x+10} ={3\over 4}$$