Question

find value of 14/(5√3-√5)​

Answer 1

Answer:

3+√5

Step-by-step explanation:

Method 1:

14+65–√=14+2×35–√

i.e. 14+65–√=9+2×35–√+5

i.e. 14+65–√=32+2×35–√+(5–√)2

i.e. 14+65–√=(3+5–√)2

Taking square root on both the sides of above equation, we get

14+65–√−−−−−−−−√=+(3+5–√) or −(3+5–√)

Method 2:

From observation we can say, 14+65–√=(a+b5–√)2

i.e. 14+65–√=a2+5b2+2ab5–√

Comparing similar terms., we get a2+5b2=14 and 2ab=6

i.e. a2+5b2=14 and ab=3

Now, consider ab=3

But, 3 has a factorization as 3=1×3

Hence, two possibilities are a=1 , b=3 or b=1 , a=3

Checking with a=1 , b=3 gives a2+5b2=12+5×32=46 which is not equal to 14

Checking with a=3 , b=1 gives a2+5b2=32+5×12=14 which is equal to 14 as per our another required condition.

Hence, a=3 , b=1 is the integer solution in 14+65–√=(a+b5–√)2

hence, 3+5–√ is the square root of 14+65–√