- WELCOME TO AN EXAMPLE ON HOW
TO SOLVE A QUADRATIC EQUATION
USING THE QUADRATIC FORMULA.
THE FIRST THING
WE SHOULD RECOGNIZE
IS THAT THE GIVEN EQUATION
IS IN THE CORRECT FORM,
WHERE THE FORM "AX" SQUARED
+ BX + C = 0.
SO WE'LL START BY IDENTIFYING
THE VALUES OF "A", B, AND C.
"A" = 9, B = 12, AND C = -1.
SO FOR THE FIRST STEP
WE'LL SUBSTITUTE THESE VALUES
INTO THE QUADRATIC FORMULA.
SO WE WOULD HAVE X = -B.
THAT'D BE -12 + OR -
THE SQUARE ROOT
OF THE QUANTITY
B SQUARED - 4AC.
THAT WOULD BE 12 SQUARED - 4
"A" IS 9 AND C IS -1.
ALL OF THIS IS DIVIDED BY 2
x "A" OR IN THIS CASE 2 x 9.
NOTICE FOR THE FIRST STEP
WE DIDN'T DO ANY SIMPLIFYING.
WE JUST PERFORMED
THE SUBSTITUTION.
NOW WE'LL BEGIN TO SIMPLIFY.
SO WE HAVE -12 + OR -
THE SQUARE ROOT.
12 SQUARED IS 144.
THEN WE HAVE - 4 x 9 x -1.
THAT'S - -36,
WHICH IS THE SAME AS + 36.
SO IT'S IMPORTANT TO TAKE
OUR TIME
WHEN SIMPLIFYING
UNDERNEATH THE SQUARE ROOT.
THIS VALUE IS CALLED
THE DISCRIMINANT.
THIS IS WHERE MANY ERRORS
CAN BE MADE.
ALL OF THIS IS DIVIDED
BY 2 x "A", IN THIS CASE 18.
SO NOW WE HAVE X = -12 + OR -
THE SQUARE ROOT
OF 144 + 36 = 180.
NOW WE WANT TO SIMPLIFY
THE SQUARE ROOT OF 180.
TO DO THIS WE'RE GOING TO FIND
THE PRIME FACTORIZATION
OF 180,
SO THAT WE CAN FIND THE
PERFECT SQUARE FACTORS OF 180.
SO 180 = 18 x 10.
18 = 9 x 2. 2 IS PRIME.
9 = 3 x 3.
THESE ARE BOTH PRIME.
AND 10 = 2 x 5, BOTH PRIME.
SO THE PRIME FACTORIZATION
OF 180 HAS TWO FACTORS OF 2,
TWO FACTORS OF 3,
AND A FACTOR OF 5.
EVERY TIME WE HAVE
TWO EQUAL FACTORS
WE HAVE A PERFECT SQUARE
FACTOR.
SO THIS WILL SIMPLIFY,
AND SO WILL THIS.
SO THE SQUARE ROOT OF 2 x 2
IS 2.
THE SQUARE ROOT OF 3 x 3 IS 3,
AND WE STILL HAVE A 5
UNDERNEATH THE SQUARE ROOT.
SO 180 SIMPLIFIES
TO 6 SQUARE ROOT 5.
SO NOW WE HAVE X = -12 + OR -
6 SQUARE ROOT 5 DIVIDED BY 18.
BUT THIS DOES SIMPLIFY,
AND THERE ARE TWO WAYS
TO SHOW THE SIMPLIFICATION.
THE WAY I RECOMMEND
IS JUST TO BREAK THIS UP
INTO TWO FRACTIONS.
SINCE WE'RE DIVIDING
BY A MONOMIAL
WE CAN WRITE THIS
AS X = - 12/18 + OR -
6 SQUARE ROOT 5/18,
AND NOW WE CAN SIMPLIFY
EACH FRACTION SEPARATELY.
12 AND 18 SHARE
A COMMON FACTOR OF 6.
THERE ARE 3 SIXES IN 18
AND 2 SIXES IN 12.
AND FOR THE SECOND FRACTION
THERE ARE 3 SIXES IN 18
AND 1 SIX IN SIX.
SO THIS SIMPLIFIES NICELY
TO X = -2/3 + OR - SQUARE ROOT
OF 5/3.
REMEMBER THIS REPRESENTS
2 SOLUTIONS.
ONE SOLUTION
IS -2/3 + SQUARE ROOT 5/3.
THE OTHER SOLUTION
IS -2/3 - SQUARE ROOT 5/3.
THE SECOND WAY
TO SIMPLIFY THIS
WOULD BE TO TAKE THIS FRACTION
HERE AND FACTOR THE NUMERATOR.
LET'S GO AHEAD AND SHOW THAT.
AGAIN, THE OTHER OPTION
IS TO WRITE X =
THERE'S A COMMON FACTOR OF 6
IN THE NUMERATOR.
SO IF WE FACTOR OUT 6,
IT WOULD LEAVE US WITH
-2 + OR - SQUARE ROOT 5/18.
WE CAN WRITE 18 AS 6 x 3.
IN THIS FORM WE CAN SEE
THERE'S A COMMON FACTOR
OF 6/6.
SO THE OTHER WAY
TO EXPRESS THE SOLUTION
AS A SINGLE FRACTION
WOULD BE X = -2
+ OR - SQUARE ROOT OF 5/3.
SO BOTH SOLUTIONS ARE CORRECT,
WRITTEN IN JUST A SLIGHTLY
DIFFERENT FORM.
ONE THING WE'VE GOT
TO BE CAREFUL ABOUT THOUGH
IS LOOKING BACK AT THIS
IN YELLOW.
WE CANNOT JUST SIMPLIFY THE 12
AND THE 18,
BECAUSE WE CANNOT SIMPLIFY
ACROSS THIS ADDITION
AND SUBTRACTION.
SO IN ORDER TO SIMPLIFY IT
WE EITHER BREAK IT UP
INTO TWO FRACTIONS,
OR FIRST FACTOR THE NUMERATOR
AND THEN SIMPLIFY
WITH THE DENOMINATOR.
I HOPE YOU FOUND THIS HELPFUL.
