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

In summary, the conversation discusses verifying whether the numbers 1 + √5 and 1 - √5 both satisfy the equation x^2 - 2x - 4 = 0. One approach is to plug the numbers into the equation and evaluate, while another is to show that the numbers have a sum of 2 and a product of -4, making them solutions of the equation. This is seen as a practice for evaluating equations.
  • #1
mathdad
1,283
1
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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  • #2
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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  • #3
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 $x^2 - 2x - 4 = 0.$
 
  • #4
This question is more evaluation practice more than anything else.
 

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

1. What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation that contains a variable raised to the power of two. It is typically written in the form of ax² + bx + c = 0, where a, b, and c are constants and x is the variable.

2. How do I verify if an equation is quadratic?

To verify if an equation is quadratic, you can check if the highest power of the variable is two. You can also rearrange the equation in the standard form of ax² + bx + c = 0 and check if the coefficients a, b, and c are real numbers.

3. What are the methods for solving quadratic equations?

The most common methods for solving quadratic equations are factoring, using the quadratic formula, and completing the square. These methods involve manipulating the equation to isolate the variable and solve for its value.

4. Can a quadratic equation have more than one solution?

Yes, a quadratic equation can have two solutions, one solution, or no real solutions at all. The number of solutions depends on the discriminant of the equation, which is the expression b² - 4ac. If the discriminant is positive, the equation will have two real solutions. If it is zero, there will be one real solution. If it is negative, there will be no real solutions.

5. How can I use the quadratic equation in real life?

The quadratic equation is used in various fields such as physics, engineering, and economics to model and solve real-life problems. For example, it can be used to calculate the maximum height of a ball thrown in the air or to determine the optimal price for a product by finding the vertex of a parabola.

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