- 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.