What Are the Conditions for a Quadratic Equation to Not Meet the X-Axis?

In summary: Happy New Year to you too! (Party)In summary, the conversation discusses a problem where the curve $y=ax^2+2bx+c$ never meets the x-axis, and the question is whether a,b,c can be in arithmetic, geometric, or harmonic progression. The summary provides a solution to show that a,b,c cannot be in arithmetic progression, and also mentions that a harmonic progression is possible. Additionally, the conversation includes a friendly exchange of New Year's greetings.
  • #1
Saitama
4,243
93
Problem:

If the curve $y=ax^2+2bx+c$, ($a,b,c \,\in\,\mathbb{R},\,a,b,c \neq 0$) never meet the x-axis, then a,b,c can't be in

A)Arithmetic Progression

B)Geometric Progression

C)Harmonic Progression

D)All of these

Attempt:

Since, the curve never meets the x-axis, we have the condition that the discriminant of the quadratic is less than zero, hence,

$$b^2<ac$$

The above shows that a,b,c can't be in geometric progression. But the given answers are A and B, how do I show that they are not in arithmetic progression? :confused:

Any help is appreciated. Thanks!
 
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  • #2
Pranav said:
Problem:

If the curve $y=ax^2+2bx+c$, ($a,b,c \,\in\,\mathbb{R},\,a,b,c \neq 0$) never meet the x-axis, then a,b,c can't be in

A)Arithmetic Progression

B)Geometric Progression

C)Harmonic Progression

D)All of these

Attempt:

Since, the curve never meets the x-axis, we have the condition that the discriminant of the quadratic is less than zero, hence,

$$b^2<ac$$

The above shows that a,b,c can't be in geometric progression. But the given answers are A and B, how do I show that they are not in arithmetic progression? :confused:

Any help is appreciated. Thanks!

Hey Pranav! ;)

If they were in arithmetic progression, you would have that:
$$b = \frac{a+c}{2}$$
Since $ac$ is greater than a square, $ac$ has to be positive, so we have:
$$\frac{a+c}{2}<\sqrt{ac}$$
According to the AM-GM inequality, this is a contradiction if $a$ and $c$ are both non-negative.
In the other case where $a$ and $c$ are both negative, we have:
$$\frac{(-a)+(-c)}{2}<\sqrt{(-a)(-c)}$$
which is again a contradiction according to the AM-GM inequality.
Therefore a,b,c can't be in arithmetic progression.

As for harmonic progression, let's try $a=1,b=\frac 1 2, c=\frac 1 3$.
That gives us:
$$\left(\frac 1 2\right)^2 < 1 \cdot \frac 1 3$$
Yep. A harmonic progression is possible. :)Oh, and even though my new year hasn't started yet, yours has.
So happy new year! (Party)
 
Last edited:
  • #3
Hi ILS! :D

I like Serena said:
If they were in arithmetic progression, you would have that:
$$b = \frac{a+c}{2}$$
Since $ac$ is greater than a square, $ac$ has to be positive, so we have:
$$\frac{a+c}{2}<\sqrt{ac}$$
According to the AM-GM inequality, this is a contradiction.
Therefore a,b,c can't be in arithmetic progression.

Ah, I was thinking about a contradiction proof. Since this is an exam problem, I wonder if it would have hit me during the exam to check the other cases after looking at $b^2<ac$.

Thanks a lot! :)

Oh, and even though my new year hasn't started yet, yours has.
So happy new year! (Party)

Yes, its been two hours since midnight. Happy New Year! :)
 
  • #4
Pranav said:
Hi ILS! :D

Ah, I was thinking about a contradiction proof. Since this is an exam problem, I wonder if it would have hit me during the exam to check the other cases after looking at $b^2<ac$.

Thanks a lot! :)

Btw, I have just edited my post, since we have to distinguish positive and negative cases.
(The AM-GM inequality has the condition that the numbers have to be non-negative.)
Yes, its been two hours since midnight. Happy New Year! :)
 
  • #5
I like Serena said:
Btw, I have just edited my post, since we have to distinguish positive and negative cases.
(The AM-GM inequality has the condition that the numbers have to be non-negative.)

Yes, thanks a lot ILS! I really need to be careful with such case-checking problems. :eek:
 

FAQ: What Are the Conditions for a Quadratic Equation to Not Meet the X-Axis?

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, meaning it has a variable raised to the power of two. It is written in the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.

What is the quadratic formula?

The quadratic formula is a method for finding the solutions to a quadratic equation. It is written as x = (-b ± √(b^2-4ac)) / 2a. It can be used for any quadratic equation, regardless of the values of a, b, and c.

How do I solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula or factor the equation. To factor, you need to find two numbers that multiply to give you the constant term (c) and add to give you the coefficient of the x term (b). These two numbers will be the factors of the quadratic equation, and you can then set each factor equal to 0 and solve for x.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is the part of the quadratic formula under the square root sign, b^2-4ac. It is used to determine the nature of the solutions to the equation. If the discriminant is positive, there will be two real solutions. If it is zero, there will be one real solution. And if it is negative, there will be two complex solutions.

How are quadratic equations used in real life?

Quadratic equations are used in various fields such as engineering, physics, and economics to model real-life situations. For example, they can be used to calculate the trajectory of a projectile, determine the optimal price for a product, or design a bridge. They are also used in computer graphics to create curved shapes and animations.

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