Understanding the Derivation of the Quadratic Formula

[sacred]

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:

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.

 

[LCKurtz]

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:

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.

 

[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.

Thank you. What about the simplification of the otherside?

 

[pwsnafu]

Do you mean the lhs? It's a perfect square. Expand it.

 

[LCKurtz]

Thank you. What about the simplification of the otherside?

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

 

[sacred]

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

 

[pwsnafu]

$\frac{b}{2a}x + \frac{b}{2a}x = \frac{1}{2}\frac{bx}{a} + \frac12\frac{bx}{a}$

 

[sacred]

$\frac{b}{2a}x + \frac{b}{2a}x = \frac{1}{2}\frac{bx}{a} + \frac12\frac{bx}{a} = \frac{bx}{a}$

Fantastic. Thank you.

 

[rama]

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

 

[SteamKing]

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