Finding the Product of Real Roots: POTW Equation Solution

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The discussion focuses on solving the equation x^2 + 18x + 30 = 2√(x^2 + 18x + 45) to find the product of its real roots. Participants confirm that the equation has no real solutions, with the correct interpretation leading to the quadratic transformation y = x^2 + 18x. The roots of the derived equation (y + 30)^2 = 4(y + 45) yield y = -20 and y = -36, leading to the valid condition y + 30 > 0. Ultimately, the product of the real roots is determined to be 20.

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Find the product of real roots of the equation ##x^2+18x+30=2\sqrt{x^2+18x+45}##.
 
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is x2 = ##x^2##?

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.
:oldconfused:
 
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malawi_glenn said:
is x2 = ##x^2##?

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.
:oldconfused:
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.
 
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anemone said:
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.:sorry:

I hope this problem looks more interesting to you now!
 
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anemone said:
I hope this problem looks more interesting to you now!
Yes, that's more like it!
 
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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}##"?
 
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julian said:
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!
 
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Coffee break solution:

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