- SO IF YOU'VE EVER WONDERED
WHERE THE QUADRATIC FORMULA CAME 
FROM, THIS IS THE VIDEO FOR YOU.
IN THIS VIDEO WE WILL DERIVE 
THE QUADRATIC FORMULA.
SO STEP ONE, WE WILL SUBTRACT C 
ON BOTH SIDES OF THIS EQUATION.
NEXT, IN ORDER TO ENSURE 
THE LEADING COEFFICIENT
IS EQUAL TO ONE, 
WE'LL DIVIDE EVERYTHING BY "A."
SO NOW WE HAVE X SQUARED 
+ B OVER "A"
x X - C OVER "A" ON THE RIGHT.
OKAY, NOW WE HAVE TO TAKE HALF 
OF THE COEFFICIENT OF THE X TERM
AND SQUARE IT.
SO LET'S GO DOWN HERE 
AND SHOW THAT WORK.
WE'RE GOING TO TAKE HALF OF B 
OVER "A"
AND THEN WE'RE GOING 
TO SQUARE THIS.
WE'LL THIS WOULD GIVE US B 
OVER 2A SQUARED
WHICH WOULD BE B SQUARED 
OVER 4A SQUARED.
THIS IS WHAT WE'RE GOING TO ADD 
TO BOTH SIDES OF THE EQUATION.
NOW THIS IS SUPPOSED TO BE 
A PERFECT SQUARE TRINOMIAL
AND IF YOU WATCHED THE PREVIOUS 
VIDEO ON COMPLETING THE SQUARE,
YOU KNOW THAT'S THE REASON WHY 
I ALWAYS LIKE TO SHOW THIS WORK.
IT MAKES FACTORING THIS 
A LOT EASIER
EVEN WHEN IT LOOKS LIKE IT'S 
GOING TO BE VERY CHALLENGING.
THIS FIRST TERM WILL BE X.
THE SECOND NUMBER OR TERM 
WILL BE WHATEVER WE SQUARED
TO GET THIS B SQUARED 
OVER 4A SQUARED.
WE TAKE A LOOK AT THE WORK.
WE SQUARED B OVER 2A TO OBTAIN 
B SQUARED OVER 4A SQUARED.
SO THIS WILL BE + B OVER 2A,
AND ON THE RIGHT SIDE WE DO HAVE 
TO ADD THESE FRACTIONS TOGETHER
SO WE'LL HAVE TO GET 
A COMMON DENOMINATOR
WHICH WOULD BE 4A SQUARED.
SO MULTIPLY THIS BY 4A OVER 4A.
SO ON THIS RIGHT SIDE
NOTICE WE DO HAVE A COMMON 
DENOMINATOR OF 4A SQUARED.
OUR NUMERATOR 
IS -4AC+ B SQUARED.
BUT I'M GOING TO WRITE THAT 
AS B SQUARED - 4AC.
OKAY, LET'S GO AHEAD AND TAKE 
THIS ONTO THE NEXT SCREEN
AND CONTINUE SOLVING.
NOW THAT WE HAVE FACTORED 
THE LEFT SIDE OF THE EQUATION,
THE NEXT STEP IS TO SQUARE ROOT 
BOTH SIDES OF THE EQUATION.
SO ON THE LEFT SIDE WE WOULD 
HAVE X + B OVER 2A,
MUST EQUAL + OR -.
NOW OUR DENOMINATOR 
IS A PERFECT SQUARE.
THE SQUARE ROOT OF 4A SQUARED 
WOULD BE 2A
AND WE'RE GOING TO LEAVE 
OUR NUMERATOR AS IS.
THE SQUARE ROOT OF B SQUARED 
- 4AC.
SO WE'RE ONE STEP AWAY NOW,
WE'LL JUST SUBTRACT B OVER 2A 
ON BOTH SIDES OF THE EQUATION
SO WE HAVE X = -B OVER 2A + OR - 
THE SQUARE ROOT
OF B SQUARED - 4AC ALL OVER 2A.
NOW THIS IS THE QUADRATIC 
FORMULA
BUT SINCE WE DO HAVE 
A COMMON DENOMINATOR,
LET'S GO AHEAD 
AND ADD THESE TOGETHER.
AND THERE WE HAVE 
THE QUADRATIC FORMULA.
SO IF WASN'T FOR COMPLETING 
THE SQUARE,
WE WOULDN'T HAVE THIS 
CONVENIENT FORMULA
TO SOLVE ANY TYPE 
OF QUADRATIC EQUATION.
SO AGAIN HERE IT IS WRITTEN 
NICELY FOR YOU.
NOW THAT WE'VE DERIVED IT 
AND KNOW WHERE IT COMES FROM,
WE SHOULD FEEL THAT MUCH BETTER 
ABOUT USING IT.
THANK YOU FOR WATCHING 
AND HAVE A GOOD DAY.
