Do 1 + √5 and 1 - √5 Solve the Equation x² - 2x - 4 = 0?

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Discussion Overview

The discussion centers on whether the numbers 1 + √5 and 1 - √5 are solutions to the quadratic equation x² - 2x - 4 = 0. Participants explore methods of verification, including direct substitution and properties of roots.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant suggests verifying the solutions by substituting 1 + √5 and 1 - √5 into the equation and checking if both yield 0.
  • Another participant agrees with the substitution method but also proposes that solving the equation and showing that the roots match the given numbers is a valid verification approach.
  • A third participant points out that the sum and product of the proposed solutions correspond to the coefficients in the quadratic equation, implying they are indeed the solutions.
  • One participant characterizes the question as primarily evaluation practice.

Areas of Agreement / Disagreement

Participants generally agree on the validity of the methods discussed for verifying the solutions, but there is no consensus on a single preferred approach.

Contextual Notes

The discussion does not resolve whether the proposed numbers are indeed solutions, as it focuses on methods of verification rather than definitive conclusions.

Who May Find This Useful

This discussion may be useful for students learning about quadratic equations and methods of verifying solutions, as well as those interested in exploring different approaches to problem-solving in mathematics.

mathdad
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Verify that the numbers 1 + √5 and 1 - √5 both satisfy the equation x^2 - 2x - 4 = 0.

I believe the question is asking to plug the given numbers into the quadratic equation and evaluate individually.

Let x = 1 + √5 and evaluate.

Let x = 1 - √5 and evaluate.

Both numbers should yield 0 = 0.

If the result for each number given is 0 = 0, then we can say that 1 + √5 and 1 - √5 are solutions of the quadratic equation.

Is this right?
 
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In my opinion, your method is valid, but I also think that solving the equation however you choose and showing the resulting roots are equivalent to the given roots is also a way to verify that they satisfy the equation. :D

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Another approach is to say that the numbers $1+\sqrt5$ and $1-\sqrt5$ have sum $2$ and product $-4$, and so they are the solutions of the equation $x^2 - 2x - 4 = 0.$
 
This question is more evaluation practice more than anything else.
 

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