Find the product of all real solutions

[anemone]

Find the product of all real solutions to the equation

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

 

[Saitama]

Find the product of all real solutions to the equation

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

Spoiler

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]

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.

Spoiler

$$y=40.5$$

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

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

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

Regards

 

[anemone]

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!

 

[CellCoree]

I would approach this problem by first analyzing the equation and its components. The equation contains four square roots, which suggests that there could be multiple solutions. I would then proceed to solve the equation by isolating the variable, y, and finding its possible values.

After solving the equation, I would obtain a set of real solutions for y. To find the product of these solutions, I would multiply all of them together. This would give me the overall product of all real solutions to the equation.

However, I would also take into consideration any restrictions or limitations on the solutions. For example, if any of the solutions make the denominator of the equation equal to zero, then those solutions would not be considered as they would result in an undefined value.

In conclusion, the product of all real solutions to the given equation would be found by solving the equation and multiplying all of the real solutions together, while also considering any restrictions on the solutions.