# Mathematics - Algebra - Quadratic Equation Formula Derivation - Step By Step

## Overview

 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

 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

   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

 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

 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

   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

 (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

 (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

 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

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