MHB How to Solve an Equation with Square Roots?

[magda21]

Please Help me solve it \[ \sqrt{x}+\sqrt{x+8}=8 \] thanks

 

[MarkFL]

Hello, and welcome to MHB! :)

I would begin by arranging as follows:

$$\sqrt{x+8}=8-\sqrt{x}$$

I really just want to arrange it so that there isn't two radicals on the same side. Now, what do you get when you square both sides?

 

[magda21]

x+8=64-x
2x=56
x=28
what I'm doing wrong?

 

[MarkFL]

Ah, you are not squaring the RHS correctly. Let's go back to:

$$\sqrt{x+8}=8-\sqrt{x}$$

Now, recall that:

$$(a+b)^2=a^2+2ab+b^2$$

And so, when we square both sides of our equation (bearing in mind that we must check for extraneous solutions), we get:

$$x+8=64-16\sqrt{x}+x$$

Collecting like terms, we can arrange this as:

$$16\sqrt{x}=56$$

Divide through by 8:

$$2\sqrt{x}=7$$

Can you proceed?

 

[magda21]

\[ \sqrt{x}=\frac{7}{2} \]
x=\frac{49}{4}
I see, thank you so much

 

[MarkFL]

Yes, and once we verifiy is works in the original equation, which it does, then we're done. :)