Proving the Quadratic Equation

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Discussion Overview

The discussion revolves around the process of deriving the quadratic formula from the standard form of a quadratic equation, ax² + bx + c = 0. Participants explore methods such as completing the square and express their understanding of the steps involved in the derivation, with a focus on both theoretical and practical aspects relevant for exams.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in understanding the transition from completing the square to the quadratic formula, specifically from the equation (x + b/2a)² = b²/4a² - c/a to the final formula.
  • Another participant suggests putting the right-hand side over a common denominator to clarify the derivation process.
  • A different approach is proposed by a participant who simplifies the problem by removing the coefficient 'a' and demonstrates the process with x² + bx + c, leading to a derived formula that they find easier to understand.
  • Some participants share their personal experiences with the formula, indicating that they find the traditional method cumbersome compared to their own derived methods.
  • There is a mention of solving a specific quadratic equation using the alternative method, with a participant questioning the validity of their approach compared to the traditional formula.
  • Another participant reflects on the advantages of the traditional method, suggesting that it may have its merits despite the preference for a simplified approach.

Areas of Agreement / Disagreement

Participants express differing views on the clarity and efficiency of various methods for deriving the quadratic formula. While some find the traditional method more cumbersome, others appreciate its structure. There is no consensus on which method is superior, as preferences vary among participants.

Contextual Notes

Some participants highlight the complexity of fractions and formulas as a barrier to understanding, while others suggest that simplifying assumptions or alternative methods may aid comprehension. The discussion reflects a range of experiences and approaches to the topic without resolving the differences in opinion.

Who May Find This Useful

This discussion may be useful for students preparing for exams, educators looking for alternative teaching methods, and anyone interested in different approaches to solving quadratic equations.

eXSBass
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Hi! I'm new here so if this has been covered please accept my apologies.
I'm having trouble proving the equation from:
ax²+bx+c=0 to the solutions x=(-b(+/-)√b²-4ac)/2a
I've been told to complete the sqaure of ax²+bx+c, then rerrange to give x, but I just can't get it!
If you could show the steps you took to get there it would be very helpful too.
Any help would be appreciated. Thanks :smile:
 
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what do you get after completing the square, assuming that you know how to complete the square? and homework ought to go in the homework section.
 
I'm sorry. Its not homework. I just need to know how for an exam. When I complete the square I get:
Well, this is what I've got on my sheet. I just don't understand step three!

ax²+bx+c=0
1 a[x²+bx+c]=0 --> x²+(b/a)x+c/a=0
2 (x+b/2a)²+c/a-b²/4a²=0 --> (x+b/2a)²=b²/4a²-c/a
3 (x+b/2a)=((+/-)√b²-4ac)/2a --> x=-b/2a((+/-)√b²-4ac)/2a
4 x=(-b(+/-)√b²-4ac)/2a

I don't understand where on the sheet it got from step 2 to step 3! That is from
(From step 2): =b²/4a²-c/a to
(From step 3): =((+/-)√b²-4ac)/2aCheers
 
Last edited:
eXSBass said:
I don't understand where on the sheet it got from step 2 to step 3! That is from
(From step 2): =b²/4a²-c/a to
(From step 3): =((+/-)√b²-4ac)/2a

Just put the RHS over a common denominator of [tex]4a^2[/tex] and then take the square-root of both sides.
 
RHS :confused: - Edit, right hand side!
 
I understand it now! Thanks a lot! It actually makes a lot more sense now! Cheers!
 
I never could follow that in high school. Now I am not sure why not, too many formulas, or fractions, i guess.

lets try it with no "a", just x^2 + bx + c. then we have to separate out the first two terms from the last:

x^2 + bx +c, then we fill in whatever makes it a square.

thats it i hate fractions, so let's begin with x^2 + 2bx +c,

then we get x^2 + 2bx +c = x^2 + 2bx + b^2 - b^2 + c

= (x+b)^2 + c - b^2. so now if it all = 0, we get (x+b)^2 = b^2 -c,

so x+b = + or - sqrt(b^2-c), and hence x = -b (+or-) sqrt(b^2-c).

now isn't that easier?

then you can solve the gbeneral case ax^2 + bx + c by changing it nito thius one:

i.e. divide by a, getting x^2 + (b/a)X + c/a. then get a 2 in there in the middle as follows:


x^2 + 2(b/2a)x + c/a. then this =0 if and only if

x = -(b/2a) (+or-) sqrt(b^2/4a^2 - c/a)

= -(b/2a) (+or-) sqrt(b^2/4a^2 - 4ac/4a^2)

= -(b/2a) (+or-) sqrt(b^2 - 4ac)/sqrt(4a^2)

= -(b/2a) (+or-) sqrt(b^2 - 4ac)/2a

= [-b (+or-) sqrt(b^2 - 4ac)]/2a.


what a messy formula, i like mine much better.
 
e.g. to solve 4x^2 + 6x -1 = 0, change it to
x^2 +(3/2)x -1/4 = x^2 + 2(3/4)x - 1/4, and the solution by my formula
is x = -(3/4) (+or-) sqrt(9/16 + 1/4) = -(3/4) (+or-)sqrt(13)/2.
is that right? it seems nicer than the usual one to me anyway.
my advice to young learners is to have somewhat less respect for tradition.
 
Last edited:
or is the usual one better? i get

x =[-6 (+or-) sqrt(36+16)]/8 i guess that's the same, and fewer fractions too! ok tradition has something going for it.
 

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