If x + 1 = x 2 and x 0, then 2x 4 is:

Question

If x + 1 = x 2 and x > 0, then 2x 4 is:

Accepted Answer

7 + 3&radic;5 — Easy Don't compute $x^4$ from the surd directly — keep reusing $x^2=x+1$ to fold powers down. Square the relation to express $x^4$ linearly in $x$, then substitute $x=	frac{1+\sqrt5}{2}$. 1 Find $x$. Rearrange $x+1=x^2$ and take the positive root: $$\begin{aligned} &x^2-x-1=0\ &\Rightarrow\ x=\frac{1+\sqrt5}{2} \quad	ext{(}x>0	ext{)} \end{aligned}$$ 2 Reduce $x^4$ to a linear expression. Square the given relation, then replace $x^2$ by $x+1$: $$\begin{aligned} &x^4=(x^2)^2=(x+1)^2=x^2+2x+1\

CAT 2017 Slot 1 — QA Question 22

Answer & solution

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