Understanding the Derivation of the Quadratic Formula

In summary, the conversation revolved around the derivation and simplification of the quadratic formula. The participant expressed a strong interest in understanding the formula and asked for an explanation. Another method for deriving the formula was also mentioned, and it was noted that the concept of quadratic equations has been around for centuries in various cultures.
  • #1
sacred
4
0
This could be seen as a rather "basic" math question, but it is a topic of curiosity for me. I'm currently a senior in high school, taking a pre-ap pre-cal/trig/AP-Calculus double blocked class. I'm absolutely fascinated by mathematics, and something of keen interest to me is the derivation of the quadratic formula. Not only do I wonder who originally derived it and how they did it, but I want to completely understand it. (Again, I can see how some people would laugh at this, because it's not that hard to understand) However, there is one part that completely blows my mind:


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How exactly does this simplify? I've sat here staring at it, attempting to conceptualize it so I can continue... but I just can't. I don't understand it. Would someone care to explain?

Thank you,


sacred


edit: reading some other threads on this board... I feel like a complete idiot... bare with me
edit2: I can understand the right side, but not the left.
 

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  • #2
sacred said:
This could be seen as a rather "basic" math question, but it is a topic of curiosity for me. I'm currently a senior in high school, taking a pre-ap pre-cal/trig/AP-Calculus double blocked class. I'm absolutely fascinated by mathematics, and something of keen interest to me is the derivation of the quadratic formula. Not only do I wonder who originally derived it and how they did it, but I want to completely understand it. (Again, I can see how some people would laugh at this, because it's not that hard to understand) However, there is one part that completely blows my mind:


596WQ.png


How exactly does this simplify? I've sat here staring at it, attempting to conceptualize it so I can continue... but I just can't. I don't understand it. Would someone care to explain?

Thank you,

sacred

In the fraction ##-\frac c a## multiply the numerator and denominator by ##4a##. That makes the two fractions on the right have the same denominator so they can be added.
 

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  • #3
LCKurtz said:
In the fraction ##-\frac c a## multiply the numerator and denominator by ##4a##. That makes the two fractions on the right have the same denominator so they can be added.

Thank you. :smile: What about the simplification of the otherside?

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  • #4
Do you mean the lhs? It's a perfect square. Expand it.
 
  • #5
sacred said:
Thank you. :smile: What about the simplification of the otherside?

597mw.png

Just square out ##\left(x+\frac b {2a}\right)^2## to see it agrees with the previous form.
 

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  • #6
LCKurtz said:
Just square out ##\left(x+\frac b {2a}\right)^2## to see it agrees with the previous form.

I did this.

I don't understand how (b/2a)x + (b/2a)x simplifies to just bx/a
 
  • #7
##\frac{b}{2a}x + \frac{b}{2a}x = \frac{1}{2}\frac{bx}{a} + \frac12\frac{bx}{a}##
 
  • #8
pwsnafu said:
##\frac{b}{2a}x + \frac{b}{2a}x = \frac{1}{2}\frac{bx}{a} + \frac12\frac{bx}{a} = \frac{bx}{a}##

Fantastic. Thank you.
 
  • #9
there is another method developed by sridhara(870-930)
ax2+bx+c=0
multiply both side by 4a
4a2x2+4abx+4ac=0
transposing 4ac
4a2x2+4abx=-4ac
add b2 to both sides
4a2x2+4abx+b2=-4ac+b2
then
(2ax+b)2=b2-4ac
2ax+b=√b2-4ac
2ax=-b(plus or minus)√(b2-4ac)
x=(-b(plus or minus)√(b2-4ac))/2a

i don't know why this is not taught in most schools,not having too many fractions this is more easier to understand since you don't take LCM
 
  • #10
The solution of quadratic equations can be traced as far back as 2000 BC in Babylonian mathematics and has developed apparently independently in several other parts of the world at much later dates.

http://en.wikipedia.org/wiki/Quadratic_equation
 

1. What is the quadratic formula?

The quadratic formula is a mathematical equation that is used to solve quadratic equations of the form ax2 + bx + c = 0, where a, b, and c are constants and x is the variable.

2. How is the quadratic formula derived?

The quadratic formula is derived using the process of completing the square on a general quadratic equation. This involves manipulating the equation to create a perfect square trinomial, which can then be easily solved for x.

3. What is the significance of the quadratic formula?

The quadratic formula is significant because it provides a universal method for solving quadratic equations, regardless of their specific form. It is also essential in many areas of mathematics and science, such as physics and engineering.

4. Can the quadratic formula be used for all quadratic equations?

Yes, the quadratic formula can be used to solve any quadratic equation, as long as the equation is in the standard form of ax2 + bx + c = 0. However, it may not always be the most efficient method for solving a particular equation.

5. Are there any alternative methods for solving quadratic equations?

Yes, there are other methods for solving quadratic equations, such as factoring, completing the square, and using the quadratic formula. The best method to use depends on the specific equation and the preferences of the individual solving it.

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