Completing the Square: Why and When?

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Completing the square is essential for graphing quadratic functions as it transforms the equation into vertex form, making it easier to visualize key features like the vertex and axis of symmetry. This method is particularly useful when dealing with quadratics that have no real roots, as it clarifies the graph's position relative to the x-axis. Additionally, mastering this technique is beneficial for future mathematical concepts, including graphing circles and solving inequalities. While it may seem unnecessary to complete the square in some cases, doing so enhances understanding and provides a clearer representation of the function. Overall, completing the square is a valuable algebraic skill that aids in both graphing and theoretical applications.
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So i just have a general knowledge question. not rly homework related

say u have a quadratic function or equation 4x2 – 2x – 5. Why would u have to complete the square in this situation? i know how to complete the square perfectly fine, but this was just bugging me.

I know that if ur graphing u complete the square to get it in the form a(x-h)^2 +c which makes it easier to visualize, but in the case of a quadratic equation i don't see why u can't just make everything equal to x as is.
 
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One completes the square to enable easy manual graphing and to help identify center, vertex, and other features of the graph.
 
Completing the square will also be used in maths later on with graphing circles, integration, inequality proofs etc.

Also, completing the square makes graphing a quadratic easier when there are no real roots.
If we solve x^2+2x+2=0 then the roots are x=-1\pm i which doesn't really tell us anything about graphing it (other than it's completely above/below the x-axis), while (x+1)^2+1 is much clearer.
 
Completing the square can be safer if you practice it, and it is good algebra practice to do it that way, if you have the time.
 
Painguy said:
So i just have a general knowledge question. not rly homework related

say u have a quadratic function or equation 4x2 – 2x – 5. Why would u have to complete the square in this situation? i know how to complete the square perfectly fine, but this was just bugging me.

I know that if ur graphing u complete the square to get it in the form a(x-h)^2 +c which makes it easier to visualize, but in the case of a quadratic equation i don't see why u can't just make everything equal to x as is.
Please note that one of the forum rules is:
In the interest of conveying ideas as clearly as possible, posts are required to show reasonable attention to written English communication standards. This includes the use of proper grammatical structure, punctuation, capitalization, and spelling. SMS messaging shorthand, such as using "u" for "you", is not acceptable.
 
vela said:
Please note that one of the forum rules is:

oops :blushing: Sorry about that. Also thanks for those who replied. I pretty much get it now.
Thanks for the help.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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