How does the quadratic formula derivation simplify?

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sacred
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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:


Two-row table of quadratic formula derivation steps: add the squared term to both sides, giving x^2 + (b/a)x + b^2/(4a^2) = -c/a + b^2/(4a^2); then simplify the right side by converting to a common denominator, giving -4ac/(4a^2) + b^2/(4a^2).

Two-row table of algebra steps: left column gives instructions, right column shows x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2) and (x + b/2a)^2 = (b^2 - 4ac)/(4a^2).



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.
 
Attachments
  • Two-row table of quadratic formula derivation steps: left column gives instructions, right column shows x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2) and (x + b/2a)^2 = (b^2 - 4ac)/(4a^2).
    597mw.png
    3.5 KB · Views: 481
  • Two-row table of steps in completing the square: adding (b/2a)^2 to both sides, giving x^2 + (b/a)x + b^2/(4a^2) = -c/a + b^2/(4a^2), then simplifying the right side to a common denominator.
    596WQ.png
    2.8 KB · Views: 515
Last edited:
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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:


Two-row table showing completing the square for a quadratic: left column text 'Add the squared term to both sides' and 'Simplify on the right-hand side', right column equations x^2 + (b/a)x + b^2/(4a^2) = -c/a + b^2/(4a^2) and x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2).


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.
 
Attachments
  • Two-row table showing completing the square for a quadratic: left column text instructions, right column equations x^2 + (b/a)x + b^2/(4a^2) = -c/a + b^2/(4a^2), then the right side simplified over a common denominator 4ac/4a^2 with b^2/4a^2.
    596WQ.png
    2.8 KB · Views: 485
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?

Two-row table of completing-the-square steps: left column gives instructions, right column shows x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2) and (x + b/2a)^2 = (b^2 - 4ac)/(4a^2).
 
Attachments
  • Two-row table of completing-the-square steps: x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2), and (x + b/(2a))^2 = (b^2 - 4ac)/(4a^2).
    597mw.png
    3.5 KB · Views: 467
sacred said:
Thank you. :smile: What about the simplification of the otherside?

Two-row table of quadratic formula steps: left column gives instructions, right column shows x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2) and (x + b/2a)^2 = (b^2 - 4ac)/(4a^2).

Just square out ##\left(x+\frac b {2a}\right)^2## to see it agrees with the previous form.
 
Attachments
  • Two-row table of quadratic formula steps: left column gives instructions, right column shows x^2 + (b/a)x + b^2/(4a^2) = -4ac/(4a^2) + b^2/(4a^2) and (x + b/(2a))^2 = (b^2 - 4ac)/(4a^2).
    597mw.png
    3.5 KB · Views: 497
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
 
##\frac{b}{2a}x + \frac{b}{2a}x = \frac{1}{2}\frac{bx}{a} + \frac12\frac{bx}{a}##
 
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.
 
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
 

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