If a = 1/(2^(1/2) - 1) - 2, what is a^2 + 2a + 4?

Question

If  a = \frac{1}{\sqrt{2}-1}-2,  what is  a^2 + 2a + 4 ?

A.  2 
 
B.  3 
 
C.  4 
 
D.  5 
 
E.  6

gmatophobia · Answer

Not sure what am I doing wrong here - a = \frac{1}{\sqrt{2}-1}-2 a = \frac{1* (\sqrt{2}+1)}{(\sqrt{2}-1) (\sqrt{2}+1)}-2 a = \frac{1* (\sqrt{2}+1)}{2-1}-2 a = (\sqrt{2}+1) -2 a + 2= (\sqrt{2}+1) -2 + 2 (a + 2)^2= (\sqrt{2}+1)^2 a^2 + 2a + 4= 3 + 2\sqrt{2}

Nabneet · Answer

a has two parts , u just cannot simplify one part of it. Its wrong
If u want to simplify then make it a single value and then u can multiply in num. & deno.

take example,
a= (1/root 2) - 2

is it same as the value , (root2/2) -2 ?

Solution:

a^2 + 2a + 4 = (a+2)^2 -2a

put the value of a in the eqn and solve

Answer we will get is 5.

Bunuel · Answer

a = \frac{1}{\sqrt{2}-1}-2= \frac{\sqrt{2}+1}{(\sqrt{2}-1)(\sqrt{2}+1)}-2=\frac{\sqrt{2}+1}{2-1}-2=(\sqrt{2}+1)-2=\sqrt{2}-1 .

a^2 + 2a + 4=(a+1)^2 + 3= (\sqrt{2}-1 + 1)^2+3=2+3=5 .

Answer: D.

gmatophobia · Answer

Thanks for your reply Nabneet . While I tried to grasp the part in which you mentioned that the rationalization part was incorrect I couldn't follow it completely. My apologies for the same. With respect to the example provided in the explanation, let me work that here with you - a = \frac{1}{\sqrt{2}} - 2 By PEDMAS rule, we first solve \frac{1}{\sqrt{2}} = \frac{1}{1.41421356237} a = 0.70710678118 - 2 = -1.29289321881 a = \frac{\sqrt{2}}{2} - 2 By PEDMAS rule, we first solve \frac{\sqrt{2}}{2} = \frac{1.41421356237}{2} a = 0.70710678118 - 2 = -1.29289321881 Aren't they the same

gmatophobia · Answer

I messed up (a+2)^2 - Where should I go and hide

Thanks Bunuel

Bunuel · Answer

Errare humanum est. It happens to the best of us.

Vaidehi06surya · Answer

hey could you tell did you come with the (a+1)sqr +3

Vaidehi06surya · Answer

hey could you tell did you come with the (a+1)sqr +3

egmat · Answer

Vaidehi06surya

When you expand  (a+1)^2 , you get:
 (a+1)^2 = a^2 + 2a + 1

But we need  a^2 + 2a + 4

So:  a^2 + 2a + 4 = (a^2 + 2a + 1) + 3 = (a+1)^2 + 3

Why This Transformation Helps:

Once we have  (a+1)^2 + 3  and know that  a = \sqrt{2} - 1 :
-  a + 1 = \sqrt{2} - 1 + 1 = \sqrt{2}  (much simpler!)
-  (a+1)^2 = (\sqrt{2})^2 = 2 
- Final answer:  2 + 3 = 5

Without this transformation, you'd have to calculate  (\sqrt{2}-1)^2 + 2(\sqrt{2}-1) + 4 , which involves expanding and simplifying multiple radical terms - much more tedious!

Strategic Pattern Recognition:

When you see expressions like  a^2 + 2a + 	ext{constant} , always check if you can complete the square:
- If the coefficient of  a  is  2 , you're looking for  (a+1)^2 = a^2 + 2a + 1 
- If the coefficient of  a  is  4 , you're looking for  (a+2)^2 = a^2 + 4a + 4 
- If the coefficient of  a  is  6 , you're looking for  (a+3)^2 = a^2 + 6a + 9

This technique saves significant time on the GMAT, especially when dealing with irrational numbers like  \sqrt{2} .

You can practice similar algebraic manipulation questions  here  (you'll find a lot of OG questions) - select  Algebra  under  Problem Solving  and choose  Medium  level questions based on your current understanding.

If a = 1/(2^(1/2) - 1) - 2, what is a^2 + 2a + 4?

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GMAT Problem Solving (PS) Questions

If a = 1/(2^(1/2) - 1) - 2, what is a^2 + 2a + 4?

Prep Toolkit

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