# Mathematics - Algebra - Quadratic Equation Formula Derivation - Step By Step ðŸ”—

## Overview ðŸ”—

 ax^2  +  bx  +  c  =  0             
 x^2  +  (bx)/a  +  c/a  =  0             
 x^2  +  (bx)/a    =  -c/a             
 x^2  +  (bx)/a  +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2           
  (x  +  b/(2a))^2   =  -c/a  +  (b/(2a))^2           
  (x  +  b/(2a))^2   =   (b^2 - 4ac)/((2a)^2)            
  x  +  b/(2a)   =   +-sqrt(b^2-4ac)/(2a)            
  x     =   (-b+-sqrt(b^2-4ac))/(2a)            

## Overview With Main Rules ðŸ”—

 ax^2  +  bx  +  c  =  0    |  x = y  =>  kx = ky  |  k  larr  1/a   
 x^2  +  (bx)/a  +  c/a  =  0    |  x = y  =>  x + k = y + k  |  k  larr  -c/a   
 x^2  +  (bx)/a    =  -c/a    |  x = y  =>  x + k = y + k  |  k  larr  (b/(2a))^2   
 x^2  +  (bx)/a  +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  k^2 +2km + m^2  =>  (k + m)^2  |  k  larr  x  m larr b/(2a)  
  (x  +  b/(2a))^2   =  -c/a  +  (b/(2a))^2  |  k / m + p / r  =>  (kr + p m) / (mr)  |  k  larr  -c  m larr a  p larr b^2  r larr (2a)^2
  (x  +  b/(2a))^2   =   (b^2 - 4ac)/((2a)^2)   |  x = y  =>  sqrt(x) = sqrt(y)       
  x  +  b/(2a)   =   +-sqrt(b^2-4ac)/(2a)   |  x = y  =>  x + k = y + k  |  k  larr  (-b/(2a))   
  x     =   (-b+-sqrt(b^2-4ac))/(2a)            

## First Step ðŸ”—

  ax^2  +  bx  +  c  =  0  |  x = y  =>  kx = ky        
 1/a  (ax^2  +  bx  +  c)  =  (1/a)0  |  k(m + p)  =>  km + kp        
  (1/a)ax^2  +  (1/a)bx  +  (1/a)c  =  (1/a)0  |  k * 0  =>  0        
  (1/a)ax^2  +  (1/a)bx  +  (1/a)c  =  0  |  (k / m)p  =>  (kp)/m        
  (1ax^2)/a  +  (1bx)/a  +  (1c)/a  =  0  |  1 * k  =>  k        
  (ax^2)/a  +  (bx)/a  +  c/a  =  0  |  (kp)/m  =>  (k / m)p        
  (a/a)x^2  +  (bx)/a  +  c/a  =  0  |  k/k  =>  1        
  1x^2  +  (bx)/a  +  c/a  =  0  |  1 * k  =>  k        
  x^2  +  (bx)/a  +  c/a  =  0            

## Next Step ðŸ”—

 x^2  +  (bx)/a  +  c/a    =  0    |  x = y  =>  x + k = y + k     
 x^2  +  (bx)/a  +  c/a  +  (-c/a)  =  0  +  (-c/a)  |  k + (-k)  =>  0     
 x^2  +  (bx)/a  +   0   =  0  +  (-c/a)  |  k + 0  =>  k     
 x^2  +  (bx)/a      =   -c/a          

## Next Step ðŸ”—

 x^2  +  (bx)/a    =  -c/a    |  x = y  =>  x + k = y + k       
 x^2  +  (bx)/a  +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2           

## Next Step ðŸ”—

  x^2  +   (bx)/a    +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  k  =>  1 * k   
  x^2  +   1((bx)/a)    +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  1  =>  k/k   
  x^2  +   (2/2)((bx)/a)    +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  (k/m)(p/r)  =>  (kp)/(mr)   
  x^2  +   (2bx)/(2a)    +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  2k  =>  k+k   
  x^2  +   (bx+bx)/(2a)    +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  (k + m)/p  =>  k/p + m/p   
  x^2  +  (bx)/(2a)  +   (bx)/(2a)  +  (b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  (km)/p  =>  k(m/p)   
  x*x  +  x(b/(2a))  +   (b/(2a))x  +  (b/(2a))(b/(2a))  =  -c/a  +  (b/(2a))^2  |  km + kp  =>  k(m + p)   
 x  (x  +  b/(2a))  +  b/(2a)  (x  +  b/(2a))  =  -c/a  +  (b/(2a))^2  |  km + kp  =>  k(m + p)   
 (x     +  b/(2a))  (x  +  b/(2a))  =  -c/a  +  (b/(2a))^2  |  k * k  =>  k^2   
 (x     +  b/(2a))^2     =  -c/a  +  (b/(2a))^2       

## Next Step ðŸ”—

 (x  +  b/(2a))^2  =  -c/a  +  (b/(2a))^2  |  (k/m)^2  =>  (k^2)/(m^2)         
 (x  +  b/(2a))^2  =  -c/a  +  (b^2)/((2a)^2)  |  k  =>  1 * k         
 (x  +  b/(2a))^2  =  1((-c)/a)  +  (b^2)/((2a)^2)  |  1  =>  k/k         
 (x  +  b/(2a))^2  =  (((2a)^2)/((2a)^2))((-c)/a)  +  (b^2)/((2a)^2)  |  (k/m)(p/r)  =>  (k/r)(p/m)         
 (x  +  b/(2a))^2  =  (((2a)^2)/a)((-c)/(2a)^2)  +  (b^2)/((2a)^2)  |  (km)^2  =>  k^2m^2         
 (x  +  b/(2a))^2  =  ((2^2a^2)/a)((-c)/(2a)^2)  +  (b^2)/((2a)^2)  |  k^2/k  =>  k         
 (x  +  b/(2a))^2  =  (4a(-c))/((2a)^2)  +  (b^2)/((2a)^2)  |  km  =>  mk         
 (x  +  b/(2a))^2  =  (-4ac)/((2a)^2)  +  (b^2)/((2a)^2)  |  k/m +p/m  =>  (k+p)/m         
 (x  +  b/(2a))^2  =   (-4ac+b^2)/((2a)^2)   |  k+m  =>  m+k         
 (x  +  b/(2a))^2  =   (b^2-4ac)/((2a)^2)              

## Next Step ðŸ”—

 (x+b/(2a))^2  =  (b^2 - 4ac)/((2a)^2)  |  x = y  =>  sqrt(x) = sqrt(y)             
 sqrt((x+b/(2a))^2)  =  sqrt((b^2-4ac)/((2a)^2))  |  sqrt(x^2)  =>  |x|             
 |x+b/(2a)|  =  sqrt((b^2-4ac)/((2a)^2))  |  |x| = y  =>  x = +-y             
 x+b/(2a)  =  +-sqrt((b^2-4ac)/((2a)^2))  |  sqrt(k/m)  =>  sqrt(k)/sqrt(m)             
 x+b/(2a)  =  +-sqrt(b^2-4ac)/sqrt((2a)^2)  |  sqrt(k^2)  =>  k             
 x+b/(2a)  =  +-sqrt(b^2-4ac)/(2a)                 

## Last Step ðŸ”—

 x  +  b/(2a)    =  +-sqrt(b^2-4ac)/(2a)    |  x = y  =>  x + k = y + k       
 x  +  b/(2a)  +  (-b/(2a))  =  +-sqrt(b^2-4ac)/(2a)  +  (-b/(2a))  |  k + (-k)  =>  0       
 x  +   0   =  +-sqrt(b^2-4ac)/(2a)  +  (-b/(2a))  |  k + 0  =>  k       
 x      =  +-sqrt(b^2-4ac)/(2a)  +  (-b/(2a))  |  k + m  =>  m + k       
 x      =  -b/(2a)  +  +-sqrt(b^2-4ac)/(2a)  |  k/m + p/m  =>  (k + p)/m       
 x      =   (-b+-sqrt(b^2-4ac))/(2a)            

## More Math ðŸ”—

Seeing Theory - A visual introduction to probability and statistics: link.

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