Can we visualize the parts of a quadratic equation as area and length?

In summary: They may even use words like 'unreal' etc., even though they can't do the maths. Generally it is a good idea to regard that sort of thing as a warning that you should try to get another opinion on the subject. I think this is a perfectly valid and intuitive way of thinking about the equation, especially for those who may struggle with the abstract concepts of algebra. However, it is important to remember that this is just an analogy and not a true representation of the equation. It can be helpful for understanding the basic concepts, but it is not always applicable to more complex equations. In summary, the equation ax^2+bx=c can be thought of as a combination of two parts, Z and Y, where Z represents
  • #1
mohamadh95
45
0
consider the equation:

ax^2+bx=c

imagine that the constant c is divided into to parts:

c=Z+Y

where ax^2= Z and bx=Y
so basically could we say c is part area and part length?? Sorry if the question is not explained better but that's the best way I came up with to describe it.
 
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  • #2
mohamadh95 said:
so basically could we say c is part area and part length??
Are you thinking about physical units?

If x is a length, then a, b and c have to have different units - as an example, a could be a number, b could be a length, and c could be an area:

$$2x^2 + (3m)x = 5m^2$$ has the solution $$x=1m$$
 
  • #3
Well I'm considering a and b to be just scalars
 
  • #4
mohamadh95 said:
Well I'm considering a and b to be just scalars
Then you would be trying to sum an area with a length. This you can't do (nor would you ever want to, anyway).

You are needing a practical example? The cost of putting in a square garden bed comprises 3 expenses:
→ a fixed cost to install a nearby tap,
→ a cost per meter for a brick border around the bed, and
→ a cost per m2 to provide a fixed depth of good soil.

Neither of the co-efficients of the variable is a scalar (without units).

If you are still unconvinced, please explain where/why you believe you would ever want to sum an area and a length? :smile:

http://imageshack.us/scaled/landing/109/holly1756.gif [Broken]
 
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  • #5
Well I'lll try to explain it with this picture
 

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  • #6
Well I'lll explain it with this picture
I'm restricted to using a tablet during this holiday period, and one of its many limitations is that it has no way to display an attachment in a way that I'm able to see the detail of a pic.

Maybe I can set it to show your pic as an inline display? Let's see ...

attachment.php?attachmentid=65132.jpg


Ah! Now that I can see!

It reveals that you are not wanting to sum an area and a length, just as I foretold. :smile: You are actually wanting to sum "pieces" with "pieces". Just as one can sum oranges with oranges, it is equally valid to sum pieces with pieces. The units of your coefficients will not be dimensionless, as we already indicated. :wink:

For your example, one coefficient will have units of pieces/metre2, the other will have units of pieces/metre. http://physicsforums.bernhardtmediall.netdna-cdn.com/images/icons/icon6.gif [Broken]
 
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  • #7
This is the only way I could explain it, I express myself better with a pen and a paper, it's a shame
 
  • #8
mohamadh95 said:
consider the equation:
ax^2+bx=c
imagine that the constant c is divided into to parts:
c=Z+Y
where ax^2= Z and bx=Y
Assumed that x is time, a is the rate of constant acceleration, and b is the initial velocity. Then Y = distance moved related to the initial velocity, and Z is the distance moved related to acceleration. C would also be the initial position at time x = 0.
 
  • #9
You can think of it with square bricks. ax2 is a squares of bricks, each side of length a, then bx is a line of b sets of bricks each of length x, or alternatively a rectangle of sides a and x, total number of bricks equals c. You can even work out solution for x whole numbers, even general solution, by moving bricks around. You can envisage a similar thing for cubic equations.

People did sometimes think about them this way. As it is hard to think about quartic and higher equations this way there were at one time many people who said - you mustn't! Such things are 'not physical', not practical, and abstract etc.

No one would say quite that now - but you may well happen to hear some of the same sort of attitude expressed, in tones of great self confidence, by marginally or non-mathematical scientists still today.
 

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, meaning it has at least one squared term. It can be written in the form of 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 mathematical formula used to solve any quadratic equation. It is written as x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the constants from the quadratic equation ax^2 + bx + c = 0.

What are the different methods for solving a quadratic equation?

There are several methods for solving a quadratic equation, including factoring, completing the square, using the quadratic formula, and graphing. The method used depends on the form of the equation and personal preference.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is the part under the square root in the quadratic formula, b^2 - 4ac. It is used to determine the nature of the solutions to the equation. If the discriminant is positive, there are two real solutions. If it is zero, there is one real solution. If it is negative, there are no real solutions.

What are the applications of quadratic equations in real life?

Quadratic equations have many real-life applications, such as in physics, engineering, and economics. They are used to model the trajectory of a thrown object, calculate the optimal shape of a bridge, and determine the profit-maximizing price for a product.

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