Help with equations

[Ximei]

I hope there is someone out there who can help me. :?
I am trying to solve this equation:
\(\displaystyle \sqrt{x+5}+\sqrt{x+3}=\sqrt{2x+7}\)
I already found out that x=>-3
I tryed to solve it and got here:
\(\displaystyle 2\sqrt{(x+5)(x+3)}= -1\)
I don't know how to solve it from here ,every method I try gives me some weird number.I’d appreciate an answer along with the steps to solve it.
And I also have to solve this one, but I don't even know from where to start
\(\displaystyle \sqrt{2x^2 -5x+2}- \sqrt{ x^2-x-2}= \sqrt{x^2-3x+2}\)
Maybe someone can give me a hint.

 

[soroban]

Hello, Ximei!

The second one is strange . . . never had one like it!

\(\displaystyle \sqrt{2x^2 -5x+2}- \sqrt{ x^2-x-2}\:=\: \sqrt{x^2-3x+2}\)

Those polynomials just happen to factor . . .


\(\displaystyle \text{We have: }\;\sqrt{(2x-1)(x-2)} - \sqrt{(x+1)(x-2)} \;=\;\sqrt{(x-1)(x-2)}\)

. . . . . . . . .\(\displaystyle \sqrt{2x-1}\!\cdot\!\sqrt{x-2} - \sqrt{x+1}\!\cdot\!\sqrt{x-2} \;=\;\sqrt{x-1}\!\cdot\!\sqrt{x-2}\)

\(\displaystyle \text{And we see that }\,\boxed{x = 2}\,\text{ is a solution.}\)



\(\displaystyle \text{If }x\neq2\text{, we can divide by }\sqrt{x-2}\!:\quad \sqrt{2x-1} - \sqrt{x+1} \;=\;\sqrt{x-1}\)

. . And this equation can be solved in the "usual way".