THIS SECTION IS ON SOLVING 
A QUADRATIC EQUATION
BY THE QUADRATIC FORMULA.
SO IN THIS SECTION,
YOU ARE TOLD THAT IF YOU HAVE
A QUADRATIC EQUATION,
AX SQUARED + BX + C = 0,
THEN THERE IS A FORMULA
THAT CAN ALLOW YOU
TO FIND THE ROOTS,
AND THAT FORMULA IS OBTAINED
BY X = -B PLUS OR MINUS
B SQUARED - 4AC ALL OVER 2A.
SO YOU CAN CONSIDER LISTENING
TO THE QUADRATIC SONG
THAT I'VE UPLOADED IT
ONTO THE WEBSITE,
OR YOU COULD CONSIDER
THIS STORY, OKAY?
THERE'S A NEGATIVE BOY
WHO COULDN'T MAKE UP HIS MIND
ABOUT GOING TO A RADICAL PARTY,
AND THE SQUARED BOY
MISSED OUT ON HIS CHANCE
WITH FOUR AWESOME CHICKS.
AND THE PARTY WAS ALL OVER
BY 2 A.M. OKAY.
SO THIS IS THE QUADRATIC STORY.
FOR SOME STUDENTS,
IT HELPS THEM REMEMBER, OKAY?
SO MAYBE YOU WANNA TRY THAT.
NOW LET ME SHOW YOU
HOW TO USE THE FORMULA.
SO SUPPOSEDLY I SAY,
"CAN YOU SOLVE
4M SQUARED = 6M + 1?"
WHAT HAPPENS IS THE VARIABLE
IN THIS QUESTION IS M.
SO VARIABLE IS M.
SO WATCH OUT FOR THAT.
UNLIKE THE ONE IN THE FORMULA
WHERE THE VARIABLE IS X.
SO BECAUSE IT'S X,
THE STORY STARTS WITH X EQUAL,
IN OUR CASE,
WE HAVE TO GO M EQUAL, OKAY?
BUT THE STORY IS THE SAME.
-B PLUS/MINUS B SQUARED
- 4AC ALL OVER 2A.
AND WHAT'S "A" AND WHAT'S B
AND WHAT'S C?
FOR YOU TO FIGURE THAT OUT,
YOU'RE GONNA HAVE TO BRING
ALL THE TERMS TO THE SAME SIDE.
SO YOU'RE GONNA HAVE TO REWRITE
AS 4M SQUARED - 6M - 1 = 0.
SO "A" IS REFERRING
TO THE NUMBER
THAT'S NEXT TO THE SQUARED TERM,
IN THIS CASE IS 4,
B IS THE NUMBER
THAT'S NEXT TO THE LETTER, OKAY?
THE ONE WITH NO EXPONENT IS -6,
AND C IS THE ONE
WITH NO VARIABLE NEXT TO IT,
SO IT'S -1.
SO SUBSTITUTE ALL THOSE VALUES
IN THERE.
SO IT'S -B, B IS -6,
PLUS/MINUS -6 SQUARED - 4 x AC.
"A" IS 4 IN THIS CASE,
AND C IS -1,
AND THE WHOLE THING IS DIVIDED
BY 2A OR 2 x 4.
SO WHAT YOU HAVE IS 6
PLUS AND MINUS SQUARE ROOT OF--
SO LET'S LOOK UP
WHAT'S IN THE RADICAL.
-6 WHEN YOU SQUARE ROOT IS 36,
AND MINUS MINUS IS PLUS 16.
SO THAT COMES UP TO BE 52.
SO WE ARE TALKING
ABOUT SQUARE ROOT 52,
AND THIS IS WHERE I HOPE
YOU STILL REMEMBER
FROM THE EARLIER CHAPTER
THAT SQUARE ROOT 52
IS REALLY 4 x 13,
WHICH REDUCES TO 2 ROOT 13.
SO YOU DO HAVE TO WRITE IT
AS 2 ROOT 13 ALL OVER 8.
BUT IF YOU STOP THERE,
YOU'RE STILL WRONG
BECAUSE YOU NOTICE
THERE'S COMMON NUMBERS
EVERYWHERE, ALL RIGHT?
THERE'S SOMETHING
THAT CAN DIVIDE THEM ALL.
SO AT THIS POINT,
YOU HAVE A SPLIT.
YOU CAN CHOOSE TO DO IT
BY FACTORING,
WHICH IS A MORE FORMAL WAY
TO DO IT,
AND THAT'S THE PROPER WAY.
TAKE OUT THE 2,
SO YOU HAVE 3 PLUS/MINUS 1
ROOT 13 ALL OVER 8,
AND THE NUMBER THAT YOU TAKE UP
BECOMES A FACTOR,
WHICH YOU CAN NOW REDUCE.
SO THE FINAL ANSWER IS GONNA BE
3 PLUS/MINUS ROOT 13/4.
SO THAT'S THE CORRECT ANSWER.
ALTERNATIVELY, YOU COULD TRY
TO GO FROM HERE TO HERE
DIRECTLY USING A SHORTCUT.
SO THE METHOD TWO IS WHERE
YOU CANCEL EACH COMMON FACTOR.
SO YOU HAVE 6 PLUS/MINUS 2
ROOT 13 OVER 8.
LOOK AT THE 6,
LOOK AT THE 2 AND LOOK AT 8.
IT'S COMMON EVERYWHERE,
AND IT CAN BE REDUCED BY 2.
SO THE 6 BECOMES A 3,
2 BECOMES A 1 AND 8 BECOMES A 4.
WHEN YOU DO THAT, YOU NOTICE
YOU CAN ALSO GET THE ANSWER.
OKAY? SO THE ONE WHERE YOU
CANCEL EACH COMMON FACTOR
IS THE FASTER WAY TO GO,
BUT MAKE SURE YOU CANCEL
FROM EVERY SINGLE FACTOR,
NOT JUST THE FRONT, OKAY?
YOU GOTTA BE FROM EVERY--
NOT JUST THE 6.
YOU GOTTA DO IT FROM THE 6,
FROM THE 2 AND FROM THE 8.
OKAY. LET'S LOOK
AT OUR SECOND EXAMPLE.
LET'S LOOK AT EXAMPLE TWO.
CAN YOU SOLVE
X SQUARED - X - 2 = 0?
SO THIS ONE,
YOU NOTICE IF YOU TRY--
IT IS ACTUALLY FACTORABLE.
IT BECOMES (X - 2)(X + 1).
AND SO WHEN YOU SET EACH FACTOR
TO 0, IT TELLS YOU X IS 2,
AND WHEN YOU SET X + 1 TO 0,
IT TELLS YOU X IS -1.
SO MY TWO ANSWERS ARE 2 AND -1,
WHICH IS A LOT FASTER, RIGHT?
BUT ALTERNATIVELY,
YOU CAN ALSO CHOOSE TO DO
BY QUADRATIC FORMULA.
SO THIS IS BY FACTORING.
OKAY, THIS IS BY FACTORING.
LET ME SHOW YOU HOW I COULD
DO IT USING THE FORMULA.
SO ASSUMING I DIDN'T MANAGE
TO FACTOR IT,
I COULDN'T FIND MY FACTORS,
SO LET ME TRY FACTORING BY--
NO, NOT BY FACTORING,
BY FORMULA.
SO "A" IS 1, B IS -1
AND THE C IS -2.
THIS TIME, MY VARIABLE
IS BACK TO X,
SO MY EQUATION IS X = -B
PLUS/MINUS B SQUARED
- 4AC ALL OVER 2A.
OKAY?
SO -B, B IN THIS CASE
IS -1, PLUS/MINUS B SQUARED.
SO YOU HAVE -1 SQUARED - 4 x AC.
"A" IS ONE AND C IS -2.
AND THE WHOLE THING
IS DIVIDED BY 2A OR 2 x 1.
SO IT'S 1 PLUS/MINUS.
LET'S DEAL WITH WHAT'S INSIDE.
-1 SQUARED
IS 1 - 4 x 1 x -2 IS PLUS--
THAT'S TWO MINUSES, RIGHT?
AND 4 x 2 IS 8.
SO THAT COMES UP TO BE 9,
MEANING ROOT 9.
SO THIS IS WHAT YOU NOTICE,
WHEN IT'S FACTORABLE,
THE P THAT'S IN THE RADICAL
IS A PERFECT SQUARE.
SO THIS P'S INSIDE THE RADICAL,
WHICH LATER ON WE HAPPEN
TO FIND OUT
THAT IT'S CALLED
THE DISCRIMINANT, OKAY?
SO THESE Ps INSIDE IS
WHAT WE CALL THE DISCRIMINANT.
WE'RE TALKING ABOUT THAT
IN A LITTLE BIT.
BUT LET'S CONTINUE.
SO ROOT 9 IS 3, DIVIDE BY 2.
SO BECAUSE
IT'S A RATIONAL NUMBER,
YOU'RE GONNA HAVE TO BREAK IT UP
AS 1 + 3/2 VERSUS 1 - 3/2.
SO THERE'S REALLY
TWO ANSWERS THERE.
SO 4/2 IS WHAT YOU CALL 2,
AND 1 - 3, THAT'S -2
DIVIDE BY 2,
WHICH COMES UP TO BE -1.
SO YOU DO HAVE THE PARAMETERS,
2M, -1
JUST LIKE WHAT WE'VE GOT
OVER HERE,
EXCEPT THAT WE KIND OF HAD
TO WORK A LITTLE HARDER.
OKAY? SO FACTORING
IS THE FASTER WAY TO GO.
THING IS, NOT EVERYTHING
IS FACTORABLE.
THAT'S WHY WE HAVE TO HAVE
THE QUADRATIC FORMULA, OKAY,
TO DEAL WITH THE ONES
THAT ARE NOT FACTORABLE.
AND LET ME SHOW YOU
OTHER EXAMPLES
LIKE, FOR EXAMPLE,
2Y SQUARED = -11 - 6Y.
SO YOU'RE GONNA COLLECT
EVERYTHING ON ONE SIDE.
YOU HAVE
2Y SQUARED + 6Y + 11 = 0.
SO THIS NEXT EXAMPLE
IS AN EQUATION
WHERE IT'S NOT FACTORABLE.
SO THIS IS WHERE YOU HAVE
TO FALL BACK
ON THE QUADRATIC FORMULA,
"A" IS 2, B IS 6 AND C IS 11.
SO I DO HAVE TO REMIND YOU
TO MAKE SURE THE VARIABLES
ARE IN DESCENDING ORDER
OF EXPONENTS. OKAY?
SO THEY HAVE TO BE
IN THE CORRECT ORDER
FOR YOU TO KNOW WHO'S "A,"
WHO'S B AND WHO'S C.
SO MAKE SURE YOU REARRANGE THEM.
AND THIS TIME,
MY VARIABLES ARE IN X.
SO VARIABLE IS IN Y--NOT X--
SORRY, Y.
SO THAT ONE OF MY EQUATION
STARTS WITH Y.
Y = -B PLUS/MINUS B SQUARED
- 4AC ALL OVER 2A.
SO THAT'S THE FORMULA.
AND SO -6 PLUS/MINUS 6 SQUARED
- 4 TIMES OF "A" AND C,
SO 2 AND 11,
ALL OVER 2A, WHICH IS 2 x 2.
SO THAT COMES UP TO BE -6
PLUS/MINUS SQUARE ROOT OF--
SO LET'S DEAL WITH THE INSIDE.
6 SQUARED IS 36 - 4 x 2.
THAT'S 8. SO IT'S MINUS 88.
SO THAT ACTUALLY COMES UP
TO BE -52.
THE DIFFERENCE
IS THE DISCRIMINANT,
WHICH IS THE NUMBER
ON THE INSIDE, OKAY,
IS NOW A NEGATIVE NUMBER.
AND THIS IS EQUIVALENT
TO ASKING YOU
TO TAKE A SQUARE ROOT OFF
A NEGATIVE NUMBER,
AND THAT'S NOT REAL.
SO THE ANSWER TO THIS QUESTION
IS NO REAL SOLUTION.
SO TELLING ME IT'S NOT
A REAL NUMBER,
IT'S NOT THE END OF THE STORY
'CAUSE THE QUESTION
IS ASKING YOU TO SOLVE.
SO MAKE SURE YOU ANSWER
AS, "NO REAL SOLUTION."
SO LET'S TRY TO SUMMARIZE HERE.
SO LET'S DO
A LITTLE SUMMARY HERE.
SO IF YOU ARE GIVEN
AX SQUARED + BX + C = 0,
SO YOU KNOW THAT THAT MEANS
THE X HAS TO BE
-B PLUS/MINUS B SQUARED - 4 = C,
ALL OVER TO 2A.
SO THE B THAT'S INSIDE,
THAT WE MENTIONED EARLIER,
SO THE 4A SQUARED - 4C,
THAT PIECE IS WHAT WE CALL
THE DISCRIMINANT.
AND WE DENOTE WITH A LETTER D.
IF YOU NOTICE
THAT IN THE LAST CASE,
YOU NOTICE WHEN B IS NEGATIVE,
THAT'S WHEN WE SAY
THAT THERE'S NO REAL SOLUTION,
RIGHT?
AND IN THE EVENT
IF D IS MORE THAN ZERO,
THERE WILL BE
TWO REAL SOLUTIONS.
MEANING, THE TWO SOLUTIONS
ARE DIFFERENT.
SO SOMETIMES WE CALL THEM
DISTINCT REAL OR DIFFERENT.
SO DIFFERENT NUMBERS,
DIFFERENT REAL NUMBERS.
OKAY?
AND WHAT HAPPENS IF D WAS ZERO?
SO EQUIVALENTLY, YOU ARE TRYING
TO TAKE THE SQUARE ROOT OF ZERO,
SO YOU END UP
WITH ONLY ONE SOLUTION.
SO THIS D TELLS YOU
A LOT OF THINGS, RIGHT?
WHEN IT'S LESS THAN ZERO IT
TELLS YOU THERE'S NO SOLUTION.
WHEN IT'S MORE THAN ZERO,
IT'S 2.
AND WHEN IT'S EQUAL TO ZERO,
IT ACTUALLY TELLS YOU
THERE'S ONE REAL SOLUTION
OR EQUIVALENTLY
WE CALL THEM AS REPEATED ROOTS.
MEANING, THE TWO SOLUTIONS
ARE REALLY THE SAME THING.
AND JUST NOW I MENTIONED TO YOU,
IF D IS A PERFECT SQUARE,
THEN THE TRINOMIAL
IS FACTORABLE.
SO IF YOU REFER
TO THE EARLIER EXAMPLE,
YOU NOTICE THERE'S ONE
WHERE WE COULD FACTOR,
AND THAT'S BECAUSE THE D
THAT WE FOUND WAS A 9.
SO A PERFECT SQUARE IS REFERRING
TO NUMBERS LIKE 1, 4, 9, 16, 25.
THOSE ARE WHAT YOU CALL
PERFECT SQUARES.
SO IF YOUR DISCRIMINANT
ARE PERFECT SQUARES,
THEN THE TRINOMIAL
IS FACTORABLE.
SO THAT'S JUST A CHECKPOINT
FOR YOU.
GREAT.
SO LET'S TRY ONE MORE EXAMPLE.
LET'S TRY EXAMPLE FOUR.
SO THIS TIME,
LET'S DEAL WITH FRACTIONS HERE.
SO SUPPOSEDLY I HAVE, OKAY--
CAN YOU SOLVE
1/4X SQUARED - 3/8 OF X = 1/4?
BUT UNLESS YOU LIKE FRACTIONS
AND CALL IT AS--SO YOU--
OF COURSE YOU HAVE
TO BRING THEM OVER FIRST.
SO IT'S
1/4X SQUARED - 3X - 1/4 = 0.
AND IF YOU LIKE,
YOU COULD PUT "A" = 1/4,
B = -3/8 AND C = -1/4.
BUT, NO, NO, NO, NO.
BAD IDEA. OKAY.
SO IT'S VERY DIFFICULT.
SO RECOMMENDED
IS TO CLEAR THE FRACTIONS.
CLEAR THE FRACTIONS,
THAT'S WHAT YOU SHOULD BE DOING.
SO WHAT YOU SHOULD DO
IS MULTIPLY THE ENTIRE EQUATION.
SO RIGHT NOW, WE HAVE
1/4X SQUARED - 3X - 1/4 = 0.
SO YOU COULD CHOOSE TO MULTIPLY
THE ENTIRE EQUATION BY 8, RIGHT?
SO YOU MULTIPLY 8 THROUGHOUT
THE EQUATION, OF COURSE,
INCLUDING THE 0.
AND SO WHAT HAPPENS
IS 8 x 1/4 BECOMES 8/4,
WHICH IS 2X SQUARED.
8 x 3/8 IS JUST 3X.
8 INTO 1/4,
THAT'S MINUS 2, RIGHT?
SO ALL OF A SUDDEN,
IT CLEARS UP, OKAY?
SO THE EQUATION
BECOMES 2X SQUARED - 3X - 2.
SO THIS IS WHERE I WAS
MENTIONING TO YOU EARLIER,
YOU COULD CHECK THE B SQUARED
- 4AC. - 4AC.
IN THIS CASE, THE "A" = 2,
THE B = -3 AND THE C = -2.
SO -3 SQUARED - 4 TIMES OF "A"
AND C.
"A" = 2 AND C = -2.
SO THAT COMES UP TO BE 9--
MINUS, MINUS--
SO THAT'S + 16 WHICH IS 25.
AHA! PERFECT SQUARE,
WHICH MEANS THAT IS FACTORABLE.
SO LET'S SEE, IS IT FACTORABLE?
SO LET'S SEE.
2X AND X,
2 + 1 MINUS AND PLUS,
SO, YES, IT'S FACTORABLE.
AND SO 2X + 1 = 0
TELLS ME X IS 2X = -1,
WHILE X = -1/2, AND X - 2 = 0
TELLS ME X = 2.
SO THAT'S TWO ANSWERS:
-1/2 OR 2.
OKAY?
TWO ANSWERS.
OR IF YOU LIKE,
YOU CAN STILL USE
QUADRATIC FORMULA.
SO THE ALTERNATIVE IS TO USE
THE QUADRATIC FORMULA.
AND SO THAT MEANS THAT X = -B,
WHICH IS -3 PLUS/MINUS
ROOT B SQUARED - 4 = C
WHICH IS 25,
SEE, I GOT IT FROM THIS ONE,
ALL OVER 2A, SO WHICH IS 2 x 2.
SO IT'S 3 PLUS/MINUS 5 OVER 4,
WHICH SPLITS INTO TWO ANSWERS.
THAT'S 3 + 5 OVER 4,
WHICH IS 8/4,
WHICH IS 2,
SO THAT'S MY 2 RIGHT THERE.
AND 3 - 5/4,
WHICH IS NEAR THE 2/4,
WHICH IS -1/2 THAT'S THERE.
SO YOU SEE THEY'RE THE SAME
TWO ANSWERS,
BUT IT'S USING FORMULA
VERSUS USING FACTORING.
