geno3141

avatar
यूजर का नामgeno3141
स्कोर17566
Stats
सवाल 1
जवाब 4450

 #1
avatar+17566 
+1

I am going to try to simplify the denominators first.

The second denominator is  sqrt(6x) - 2·sqrt(3).

However, 2·sqrt(3) can be written as sqrt(4)·sqrt(3) = sqrt(12)

     and sqrt(12) can be written as sqrt(6)·sqrt(2),

so sqrt(6x) - 2·sqrt(3) becomes sqrt(6)·sqrt(x) - sqrt(6)·sqrt(2),

and, factoring, becomes:  sqrt(6)[sqrt(x) - sqrt(2)].

This is almost the same as the first denominator, which is simply [sqrt(x) - sqrt(2)].

 

This means that if x = 2, these denominators are zero, so if we get an answer of 2, we'll have to throw it out.

 

Now, let's multiply both sides by sqrt(6)[sqrt(x) - sqrt(2)].

This cancels the denominator on the left side of the equation, but puts a factor of sqrt(6) on the left side.

This completely cancels the denominator on the right side of the equation.

 

We now have:  sqrt(6)·[sqrt(3x) - 4sqrt(3)]  =  2sqrt(2x) + sqrt(2)

 

Let's square both sides:

 

Left side:     [ sqrt(6)·[sqrt(3x) - 4sqrt(3)] ]2  = 6·[ sqrt(3x) - 4sqrt(3) ]2  =  6[ 3x - 2·4·sqrt(3x)·sqrt(3) + 16·3 ]

     =  6[3x - 8·3·sqrt(x) + 48]  =  18x - 144sqrt(x) + 288

 

Right side:     [ 2sqrt(2x) + sqrt(2) ]2  =  4·2x + 2·2·2·sqrt(x) + 2  

     =  8x + 8sqrt(x) + 2

 

Setting these two equal:     18x - 144sqrt(x) + 288  =  8x + 8sqrt(x) + 2

Simplifying:     10x - 152sqrt(x) + 286 = 0

Dividing by 2:     5x - 76sqrt(x) + 143  =  0

 

Using the quadratic formula to find sqrt(x);     sqrt(x)  =  [ 76 +/1 sqrt(762 - 4·5·143) ] / 10

----->     sqrt(x)  =  13 or 2.2

----->     x = 169  or  x = 4.84

 

Checking both possible answers, 169 works but 4.84 is an extraneous root introduced by the squaring process.

geno3141 2017 Sep 17