What is the value of (a*5^(1/2) + b*5^(1/2))^2?

Question

What is the value of  (a\sqrt{5} + b\sqrt{5})^2 ?

(1) a - b = 5
(2) a(a + b) = 81 - b(b + a)

NiharikaR · Answer

IMO B
5(a-b)^2=?

St1
a-b=5
doesn't help

St2
(a+b)^2=81
Hence sufficient

DavidTutorexamPAL · Answer

As all we're given is equations, we'll look for a simplification-based approach.
This is a Precise methodology.

We'll simplify the data in our question stem:
 (a\sqrt{5} + b\sqrt{5})^2=(\sqrt{5}(a+b))^2=5(a+b)^2

(1) This lets us substitute b instead of a into the equation which is not enough to calculate its value.
Insufficient.

(2) Moving -b(b+a) to the LHS and extracting (a+b) as common factor gives (a+b)(a+b)=81 so (a+b)^2 = 81
Then our answer is 5*81.
Sufficient.

(B) is our answer.

EgmatQuantExpert · Answer

Solution

Given: 
 • We are given an expression  (a \sqrt{5}+b \sqrt{5}) ^2

To find:

• We need to find the value of the expression  (a \sqrt{5}+b \sqrt{5}) ^2 .
 o  (a \sqrt{5}+b \sqrt{5}) ^2  =    ^2 
o =  5(a + b) ^2

Thus, we only need to find the value of (a + b) to find the value of the expression  (a \sqrt{5}+b \sqrt{5}) ^2 .

Statement-1:  “ a - b =5 “

The value of a-b can be 5 for various values of a and b. And, different values of a and b will result in a different value of a + b.

Let us see some examples:
 • For, a=6 and b=1, a-b=5 and a + b = 7
• For, a=7 and b=2, a-b=5 and a + b = 9

Thus,  Statement 1 alone is not sufficient to answer the question .

Statement-2:  “ a (a + b) = 81- b (a + b)  “

Let us simplify the expression “a (a + b) = 81- b (a + b)”.
 •  a^2+ ab = 81- b^2-ab 
•  a^2+b^2+2ab= 81 
•  (a + b) ^2 = 9^2 
•  a + b= 9

We can find the value of (a + b) from statement 2.  Hence, we can also find the value of  (a \sqrt{5}+b \sqrt{5}) ^2 .

Thus,  Statement 2 alone is sufficient to answer the question .

Hence, the correct answer is option B.

Answer: B

What is the value of (a5^(1/2) + b5^(1/2))^2?