sqrt(x-27) + sqrt(x) = 9

solve for x...

i tried squaring both sides to get the sqrts to go away but i still couldnt get the right answer.

I need to know how to get to the final solution. FYI the answer is 36.

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- Sep 11th 2008, 10:21 AMjohntuanAlgebraic Manipulations and Equations
sqrt(x-27) + sqrt(x) = 9

solve for x...

i tried squaring both sides to get the sqrts to go away but i still couldnt get the right answer.

I need to know how to get to the final solution. FYI the answer is 36. - Sep 11th 2008, 10:29 AMMoo
Hello,

Square it :

$\displaystyle (x-27)+x+2 \sqrt{x(x-27)}=81$

$\displaystyle 2x+2 \sqrt{x(x-27)}=108$

$\displaystyle \sqrt{x(x-27)}=54-x$

Square again and solve the quadratic :)

Recall that $\displaystyle x-27 \ge 0$ and $\displaystyle x \ge 0$ (which makes $\displaystyle x \ge 27$) - Sep 11th 2008, 11:04 AMMoo
Another way of doing it :

Let $\displaystyle a=\sqrt{x-27}$ and $\displaystyle b=\sqrt{x}$

We have $\displaystyle a+b=9$

Now, multiply both sides by $\displaystyle (a-b)$ :

$\displaystyle \underbrace{(a+b)(a-b)}_{a^2-b^2}=9(a-b)$

$\displaystyle (x-27)-x=9(a-b)$

$\displaystyle -27=9(a-b)$

$\displaystyle a-b=-3$

So you have :

$\displaystyle \left\{\begin{array}{ll} a+b=9 \quad (1) \\ a-b=-3 \quad (2) \end{array} \right.$

$\displaystyle (1)-(2) ~:~ 2b=12 \implies b=6 \implies \sqrt{x}=6 \implies x=36$

Check if it is correct :

$\displaystyle (1)+(2) ~:~ 2a=6 \implies a=3 \implies \sqrt{x-27}=3 \implies x-27=9 \implies x=36$

Yay ! - Sep 11th 2008, 11:42 AMshinhidora
- Sep 11th 2008, 11:48 AMMoo
- Sep 11th 2008, 11:50 AMmasters
Lookie here! You might like it this way, too. Pardon me, Moo. I just needed to post something. Haven't been on a lot today.

$\displaystyle \sqrt{x-27}+\sqrt{x}=9$ Move the $\displaystyle \sqrt{x}$ to the RHS.

$\displaystyle \sqrt{x-27}=9-\sqrt{x}$ Square both sides.

$\displaystyle x-27=81-18\sqrt{x}+x$ Collect terms.

$\displaystyle -108=-18\sqrt{x}$ Square both sides again.

$\displaystyle 11664=324x$ Divide

$\displaystyle \boxed{x=36}$ - Sep 11th 2008, 11:53 AMshinhidora