+0

# I would like to know the solution of the following equation.

0
138
2

I would like to know the solution of the following equation:

$$\sqrt{a-\sqrt{a+x}} = x$$

Hamed

Guest Mar 2, 2015

#1
+91436
+5

What do you mean by solution.  there are 2 unknowns  Do you want the integer solutions?

I'll try to make x the subject

$$\begin{array}{rll} \sqrt{a-\sqrt{a-x}}&=&x\\\\ a-\sqrt{a-x}&=&x^2\\\\ -\sqrt{a-x}&=&x^2-a\\\\ a-x&=&(x^2-a)^2\\\\ a-x&=&x^4-2ax^2+a^2\\\\ a-a^2-x+2ax^2-x^4&=&0\\\\ -((a-x-x^2)(-1+a+x-x^2))&=&0\\\\ (a-x-x^2)(-1+a+x-x^2)&=&0\\\\ (a-x-x^2)(-1+a+x-x^2)&=&0\\\\ \end{array} so \\ a-x-x^2=0\qquad or \qquad -1+a+x-x^2=0\\\\ x^2+x-a=0\qquad or \qquad x^2-x+(1-a)=0\\\\$$

You can now solve these seperately using the quadratic formula and you will end up with 4 solutions where x is the subject.

the numeric value of x will depend on your value of a.

here is a graph

https://www.desmos.com/calculator/oxyumtk0dq

Melody  Mar 3, 2015
Sort:

#1
+91436
+5

What do you mean by solution.  there are 2 unknowns  Do you want the integer solutions?

I'll try to make x the subject

$$\begin{array}{rll} \sqrt{a-\sqrt{a-x}}&=&x\\\\ a-\sqrt{a-x}&=&x^2\\\\ -\sqrt{a-x}&=&x^2-a\\\\ a-x&=&(x^2-a)^2\\\\ a-x&=&x^4-2ax^2+a^2\\\\ a-a^2-x+2ax^2-x^4&=&0\\\\ -((a-x-x^2)(-1+a+x-x^2))&=&0\\\\ (a-x-x^2)(-1+a+x-x^2)&=&0\\\\ (a-x-x^2)(-1+a+x-x^2)&=&0\\\\ \end{array} so \\ a-x-x^2=0\qquad or \qquad -1+a+x-x^2=0\\\\ x^2+x-a=0\qquad or \qquad x^2-x+(1-a)=0\\\\$$

You can now solve these seperately using the quadratic formula and you will end up with 4 solutions where x is the subject.

the numeric value of x will depend on your value of a.

here is a graph

https://www.desmos.com/calculator/oxyumtk0dq

Melody  Mar 3, 2015
#2
+91436
0

I cheated a little by getting Wolfram Alpha to do this factorising step for me.

Can anyone show me how to do it by hand?    Thank you

$$a-a^2-x+2ax^2-x^4&=&0\\\\ -((a-x-x^2)(-1+a+x-x^2))&=&0\\\\$$

Melody  Mar 3, 2015

### 11 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details