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:

A.  6 + 4√5

B.  3 + 5√5

C.  7 + 3√5

D.  5 + 3√5

E.  4 + 3√5

Are You Up For the Challenge: 700 Level Questions

andreagonzalez2k · Accepted Answer

Solving  x^2-x-1=0  the positive root is  x=\frac{1+\sqrt{5}}{2}

Playing a little with the expression:
 x^2=x+1 
 x^4=(x+1)^2=x^2+2x+1=x+1+2x+1=3x+2 
 2x^4=6x+4

Substituting:
 6*\frac{1+\sqrt{5}}{2}+4=3+3\sqrt{5}+4=7+3\sqrt{5}

IMO C

chetan2u · Answer

x^2-x-1=0…….x^2-x+(\frac{1}{2})^2-(\frac{1}{2})^2-1=0….(x-\frac{1}{2})^2-\frac{5}{4}=0

(x-\frac{1}{2}-\frac{\sqrt{5}}{2}) (x-\frac{1}{2}+\frac{\sqrt{5}}{2})=0

As x>0,  x=\frac{1}{2}+\frac{\sqrt{5}}{2} =\frac{1+\sqrt{5}}{2}

Thus,  x^2=\frac{(1+5+2\sqrt{5})}{4}=\frac{3+\sqrt{5}}{2}

=>  x^4=\frac{(9+5+6\sqrt{5})}{4}=\frac{7+3\sqrt{5}}{2}

And,  2x^4=2*\frac{7+3\sqrt{5}}{2}=7+3\sqrt{5}

wtstone · Answer

could you elbaorate or how you get (1/2)^2? I am following the rest, but I am hung-up at the start. explaining that would really help me out.

Oppenheimer1945 · Answer

The most loved sequence you’ll find in the nature (from sunflower seed to cyclones to the the structures of galaxies). x^2=x+1 or T_n=T_(n-1)+T_(n-2).

I was first introduced this concept while solving advanced level problem on Limits of calculus.

If you know the solution of the quadratic is the golden ratio of 1.6 .

Golden Ratio is one of the most common mathematical ratios in nature. We see this ratio everywhere from majestic landscapes like the Pyramids of Giza and the Mona Lisa to modern-day logos such as Twitter and Pepsi. Golden ratios are unique because of their golden proportion.

2x^4=2*(1.6)^4=close to 13.

Or x^4=3x+2 giving C as ans.

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions