Finding the Product of Real Roots: POTW Equation Solution

[anemone]

Find the product of real roots of the equation $x^2+18x+30=2\sqrt{x^2+18x+45}$.

 

[malawi_glenn]

is x2 = $x^2$?

Spoiler

Find the product of real roots of the equation x^2+18x+30=2x^2+18x+45.
I would say that the equation has no real roots hence the product of real roots is not defined.
Unless I have missed something REALLY obvious or some technicality.

 

[Mark44]

is x2 = $x^2$?

Spoiler

Find the product of real roots of the equation x^2+18x+30=2x^22+18x+45.
I would say that the equation has no real roots hence the product of real roots is not defined.
Unless I have missed something REALLY obvious or some technicality.

My best guess is that x2 was indeed meant to represent $x^2$.
In your spoiler work, it looks like you have an extra '2' in the exponent on the right side.
I also agree that the original equation (with my assumption) has no real solutions.

 

[anemone]

Find the product of real roots of the equation $x^2+18x+30=2\sqrt{x^2+18x+45}$.

Hello again to all!

Let me thank first for the moderator who edited my post to fix the caret sign.

But, this is actually the intended equation! I didn't miss out the square root but somehow I guess I did. Sorry.

I hope this problem looks more interesting to you now!

 

[julian]

I hope this problem looks more interesting to you now!

Yes, that's more like it!

 

[julian]

Hello @anemone. Welcome to the PF!

I just want to check. If it is a high school maths problem then should the question be: "Find the product of real roots of the equation $x^2 + 18 x + 30 = 2 \sqrt{x^2 + 18 x + 45}$"?

 

[anemone]

Hello @anemone. Welcome to the PF!

I just want to check. If it is a high school maths problem then should the question be: "Find the product of real roots of the equation $x^2 + 18 x + 30 = 2 \sqrt{x^2 + 18 x + 45}$"?

Thanks so much @julian, post is edited!

 

[malawi_glenn]

Coffee break solution:

Spoiler

square both sides
$(x^2+18x+30)^2 = 4(x^2+18x+45) = 4(x^2+18x+30)+60$
define $t=x^2+18x+30$
$t^2 -4t - 60 =0$
We seek solutions s.t. $t>0$ to solve the original equation, we obtain
$t=10$ ($t=-6$ not valid, false root).
We thus have $x^2+18x+30 = 10$ i.e. $x^2+18x+20= 0$.
In the equation $x^2 + Ax+B=0$ the product of the real roots is equal to $B$,
hence the product of the real roots for the original equation is 20.

 

[anemone]

Congratulations to @malawi_glenn for your correct solution! And thanks for your participation!

 

[anuttarasammyak]

$$x^2+18x:=y$$
$$(y+30)^2=4(y+45)$$
the solution of the quadratic equation is y=-20,-36. we have the condition
$$y+30>0$$
so y=-20
$$x^2+18x+20=0$$
The product of the roots is 20.