# Find the product of all real solutions

#### anemone

##### MHB POTW Director
Staff member
Find the product of all real solutions to the equation

$\sqrt{2y-81}-\sqrt{1734-20y+4y^2}-\sqrt{81-2y}+\sqrt{4y^2+26y-129}=0$.

#### Pranav

##### Well-known member
Find the product of all real solutions to the equation

$\sqrt{2y-81}-\sqrt{1734-20y+4y^2}-\sqrt{81-2y}+\sqrt{4y^2+26y-129}=0$.

From the first surd, we have the condition $y \geqslant \frac{81}{2}$.

From the third surd, we have the condition $y \leqslant \frac{81}{2}$. But from the first condition, we have only one possible value of y for which both these expressions are defined i.e $\frac{81}{2}$

Substituting 81/2 in the given equation makes the first and the third surd equal to zero. Now we have to check if the remaining two surds give the same value when y=81/2 is substituted. Directly substituting 81/2 takes time (at least for me) so we take the reverse path. We equate the expressions inside the remaining surds to see at what value of y they are equal, if y comes out to be 81/2, then 81/2 is the answer.

$$4y^2+26y-129=4y^2-20y+1734 \Rightarrow 46y=1863 \Rightarrow y=\frac{81}{2}$$

Hence, the answer is $\boxed{\dfrac{81}{2}}$

#### mente oscura

##### Well-known member
Find the product of all real solutions to the equation

$\sqrt{2y-81}-\sqrt{1734-20y+4y^2}-\sqrt{81-2y}+\sqrt{4y^2+26y-129}=0$.
Hello.

Has you healed?. I'd like to that you have recovered.

$$y=40.5$$

$$\forall{y}>40.5 \ and \ \forall{y}<40.5 \ \rightarrow{}$$

$$\rightarrow{} \sqrt{2y-81}-\sqrt{1734-20y+4y^2}-\sqrt{81-2y}+\sqrt{4y^2+26y-129} \neq{0}$$

$$key \ term=\sqrt{81 \pm{2y}}$$

Regards

#### anemone

##### MHB POTW Director
Staff member
Thanks to both of you for participating and you have gotten the right answer! Well done!

Hello.

Has you healed?. I'd like to that you have recovered.
Yes, I have recovered fully from food poisoning and thanks for asking!