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# halp

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What is the sum of the solutions to the following equation? $$\sqrt{x}=\frac{12}{7-\sqrt{x}}$$.

Dec 9, 2018

#1
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It should be $$81+256=337.$$ First, change the fourth root into $$\frac{1}{4}.$$

Now, without words:

$$x^{\frac{1}{4}}=\frac{12}{7-x^{\frac{1}{4}}}$$

$$x^{\frac{1}{4}}\left(7-x^{\frac{1}{4}}\right)=\frac{12}{7-x^{\frac{1}{4}}}\left(7-x^{\frac{1}{4}}\right)$$

$$x^{\frac{1}{4}}\left(7-x^{\frac{1}{4}}\right)=12$$

Expand, $$7x^{\frac{1}{4}}-\sqrt{x}=12$$

$$7x^{\frac{1}{4}}-\left(x^{\frac{1}{4}}\right)^2=12$$

Plug variables instead of $$x^\frac{1}{4}$$

to get $$x=81,\:x=256$$

Thus, te answer is $$81+256=\boxed{337}.$$

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Dec 9, 2018
#2
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Let's see hoe tertre arrived at his solution

7x^(1/4) - ( x^1/4)^2 = 12     multiply through by -1

(x^1/4)^2 - 7(x^1/4) = -12

(x^1/4)^2 - 7(x^1/4) + 12 = 0         let  x^1/4 = a     and we have

a^2 - 7a + 12 = 0     factor

(a - 4) (a - 3) = 0

Setting both factors to 0 and solving for "a" gives   a =4   or a = 3

So

x^1/4 = 4         or       x^1/4 = 3      take each side to the 4th power

x = 256          or       x =  81

And the sum of these is 337.....just as tertre found !!!!!   Dec 9, 2018