Finding the Product of Real Roots: POTW Equation Solution

In summary, to find the product of real roots in the POTW equation, you can use the quadratic formula, factor the equation into its roots and multiply them together, or use Vieta's formulas. The POTW equation can have multiple real roots, with the number depending on the discriminant. If the discriminant is greater than zero, there are two real roots. If it is equal to zero, there is only one real root. And if it is less than zero, there are no real roots and the equation has complex roots.
  • #1
anemone
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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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  • #2
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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  • #3
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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  • #4
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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  • #5
anemone said:
I hope this problem looks more interesting to you now!
Yes, that's more like it!
 
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  • #6
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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  • #7
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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  • #8
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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  • #9
Congratulations to @malawi_glenn for your correct solution! And thanks for your participation!
 
  • #10
[tex]x^2+18x:=y[/tex]
[tex](y+30)^2=4(y+45)[/tex]
the solution of the quadratic equation is y=-20,-36. we have the condition
[tex]y+30>0[/tex]
so y=-20
[tex]x^2+18x+20=0[/tex]
The product of the roots is 20.
 
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FAQ: Finding the Product of Real Roots: POTW Equation Solution

What is the POTW equation?

The POTW equation is a mathematical equation that represents a polynomial with real coefficients. It stands for "Product of the Two Roots of a Polynomial".

How do I solve the POTW equation?

To solve the POTW equation, you need to first find the roots of the polynomial. This can be done by factoring the polynomial or by using the quadratic formula. Once you have the roots, simply multiply them together to find the product.

Why is finding the product of real roots important?

Finding the product of real roots is important because it gives us information about the behavior of the polynomial. The product of the roots can tell us whether the polynomial has a positive or negative leading coefficient, and whether the graph of the polynomial will have a positive or negative end behavior.

Can the POTW equation be used for polynomials with complex roots?

No, the POTW equation can only be used for polynomials with real roots. Complex roots cannot be multiplied together to find a product.

Are there any other methods for finding the product of real roots?

Yes, there are other methods such as using the Vieta's formulas or the Descartes' rule of signs. However, these methods are more complex and are not as commonly used as factoring or the quadratic formula.

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