Finding the Product of Real Roots: POTW Equation Solution

  • Context: High School 
  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Product Roots
Click For Summary

Discussion Overview

The discussion revolves around finding the product of the real roots of the equation ##x^2+18x+30=2\sqrt{x^2+18x+45##. Participants explore the equation's structure, potential solutions, and the nature of its roots, with a focus on whether real solutions exist.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants question the notation used in the equation, specifically whether ##x^2## was correctly represented.
  • One participant suggests that the original equation, based on their assumptions, has no real solutions.
  • A later post clarifies that the equation presented is indeed the intended one, correcting earlier misunderstandings about the square root.
  • Another participant proposes a method to solve the equation by substituting ##y = x^2 + 18x##, leading to a quadratic equation.
  • One participant claims to have found the product of the roots as 20, based on their calculations.

Areas of Agreement / Disagreement

Participants express differing views on the existence of real solutions, with some suggesting there are none while others provide a method that leads to a solution. The discussion remains unresolved regarding the overall validity of the proposed solutions.

Contextual Notes

There are unresolved assumptions regarding the conditions under which the roots are considered real, as well as the implications of the transformations applied to the original equation.

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Find the product of real roots of the equation ##x^2+18x+30=2\sqrt{x^2+18x+45}##.
 
Last edited:
  • Like
  • Love
Likes   Reactions: Euge, malawi_glenn, cbarker1 and 3 others
Physics news on Phys.org
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:
 
Last edited:
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.
 
  • Like
Likes   Reactions: malawi_glenn and jim mcnamara
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!
 
Last edited:
  • Like
Likes   Reactions: malawi_glenn
anemone said:
I hope this problem looks more interesting to you now!
Yes, that's more like it!
 
  • Like
Likes   Reactions: malawi_glenn
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}##"?
 
Last edited:
  • Like
Likes   Reactions: anemone and malawi_glenn
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!
 
  • Like
Likes   Reactions: julian and malawi_glenn
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.
 
Last edited:
  • Like
Likes   Reactions: PeroK and anemone
Congratulations to @malawi_glenn for your correct solution! And thanks for your participation!
 
  • #10
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.
 
  • Like
Likes   Reactions: Office_Shredder

Similar threads

High School Is It Possible to Prove Real Root Properties in Cubic Equations?
  • · Replies 1 ·
Replies
1
Views
2K
High School How Can One Non-Real Root Be Guaranteed in Equation #416?
  • · Replies 1 ·
Replies
1
Views
2K
High School Maximize (x1+x2)(x1+x3)x4 for Quartic Equation with Real Roots in [1/2,2]
  • · Replies 2 ·
Replies
2
Views
2K
High School Can Non-zero Polynomials with Real Coefficients Have Real Roots in Common?
  • · Replies 1 ·
Replies
1
Views
2K
High School What Is the Value of b If All Roots of the Polynomial Are Positive Real Numbers?
  • · Replies 2 ·
Replies
2
Views
2K
High School How to Calculate \(a+b+c\) for Equations Sharing Common Real Roots?
  • · Replies 1 ·
Replies
1
Views
1K
High School How Can We Prove the Inequality for Polynomial Roots in POTW #394?
  • · Replies 1 ·
Replies
1
Views
2K
High School Can Real Roots in a Quadratic Guarantee a Root in a Quartic?
  • · Replies 1 ·
Replies
1
Views
2K
High School How Do You Evaluate the Product of g(x) at the Roots of a Quintic Polynomial?
  • · Replies 1 ·
Replies
1
Views
2K
High School Prove: No Real Roots for Polynomial x^4-px^3+qx^2-sqrt(q)x+1
  • · Replies 2 ·
Replies
2
Views
2K