- IN THIS VIDEO, WE'RE GOING
TO LEARN HOW TO SOLVE
ANY QUADRATIC EQUATION
USING THE QUADRATIC FORMULA.
SO WHAT IS THE QUADRATIC
FORMULA?
WELL, IF YOU HAVE AN EQUATION
THAT'S QUADRATIC,
REMEMBER,
ALL QUADRATIC EQUATIONS
CAN BE WRITTEN LIKE THIS,
THEN, THE SOLUTIONS ARE
X = -B + OR - THE SQUARE ROOT
OF B SQUARED - 4AC ALL OVER 2A.
THIS IS A FORMULA CALLED
THE QUADRATIC FORMULA.
THIS FORMULA MUST, MUST, MUST,
MUST BE MEMORIZED,
WRITE IT DOWN A HUNDRED TIMES,
DO WHAT EVER YOU HAVE TO DO
IN ORDER TO REMEMBER IT.
SOME COMMON MISTAKES ARE THAT,
SOMETIMES STUDENTS DON'T PUT
THE WHOLE THING OVER 2A.
SO MAYBE THEY'LL JUST PUT THIS
PART OVER 2A, AND NOT THE -B,
SO THIS WOULD BE WRONG.
SOMETIMES, THEY FORGET THE -B,
RIGHT.
SOMETIMES, THEY JUST PUT B.
THERE ARE A LOT OF WAYS THAT
STUDENTS CAN MAKE A MISTAKE
HERE. OKAY.
YOU'VE GOT TO PRACTICE WRITING
THIS DOWN,
PUT IT ON A FLASH CARD
AND JUST MEMORIZE IT.
NOW, THIS IS A VERY USEFUL
FORMULA,
WE CAN SOLVE ANY QUADRATIC
EQUATION USING THIS FORMULA
SO WE DON'T NEED TO USE
FACTORING ANYMORE,
WE DON'T NEED TO USE
COMPLETING THE SQUARE,
WE DON'T NEED TO USE
THE SQUARE ROOT PROPERTY,
THIS FORMULA GIVES US
THE ANSWERS WE WANT. OKAY.
NOW, THAT DOESN'T MEAN THAT
YOU DON'T EVER, EVER WANT TO USE
FACTORING.
FACTORING IS STILL THE EASIEST
METHOD THAT WE HAVE. OKAY.
WHAT THIS MEANS, THOUGH, IS
IF SOMETHING CAN NOT FACTOR,
THEN, ALWAYS GO FOR
THE QUADRATIC EQUATION. OKAY.
SO LETS PRACTICE USING
THE EQUATION.
WHAT ARE THE STEPS FIRST?
SO YES, THERE ARE A FEW STEPS
YOU SHOULD GO THROUGH FIRST
TO MAKE YOUR LIFE EASIER.
SO STEP ONE,
STEP ONE IS ALWAYS GOING TO BE
TO PUT IN STANDARD FORM,
REMEMBER, STANDARD FORM IS
AX SQUARED + BX + C = 0.
YOU HAVE TO GET IT ALL EQUAL
TO ZERO
AND YOU HAVE TO PUT IT
IN DESCENDING ORDER.
STEP TWO, ALSO,
CLEAR ALL FRACTIONS
BY FIRST, MULTIPLYING
YOUR EQUATION BY THE LCD.
NOW, YOU DON'T USUALLY HAVE
FRACTIONS BUT WHEN YOU DO,
THAT'S HOW YOU'RE GOING
TO GET RID OFF THEM,
YOU DON'T WANT TO STICK
FRACTIONS INTO THAT FORMULA.
TRUST ME, IT'S GOING TO GET
REALLY UGLY, REALLY FAST.
STEP THREE, IDENTIFY A, B, AND C
AND THEN, PLUG INTO FORMULA
AND SIMPLIFY.
I'D SUGGEST THREE STEPS AND THE
FIRST TWO ARE PRELIMINARY,
YOU DO THOSE BEFORE YOU EVEN USE
THE FORMULA. ALL RIGHT.
SO LET'S GET GOING, LET'S TRY,
FOR ALL OF THESE, BY THE WAY,
WE'RE GOING TO SOLVE USING
THE QUADRATIC FORMULA.
SO PROBLEM ONE, 5P SQUARED - P
- 3 = 0.
ALL RIGHT, SO FIRST OF ALL,
THIS IS ALREADY IN STANDARD FORM
BECAUSE EVERYTHING'S ON THE LEFT
AND IT'S IN DESCENDING ORDER,
SO STEP ONE IS DONE FOR US
ALREADY.
STEP TWO, THERE ARE NO FRACTIONS
WE CAN SKIP THAT ONE.
STEP THREE, IDENTIFY A, B,
AND C.
SO HERE, A IS 5, B IS -1,
AND C IS -3.
I WROTE DOWN ALL MY COEFFICIENTS
SO NOW I USE THE FORMULA.
REMEMBER, X IS -B, + OR - THE
SQUARE ROOT OF B SQUARED - 4AC
ALL OVER 2A,
WHICH IS -B + OR - THE SQUARE
ROOT OF B SQUARED
MINUS 4 x A x C ALL OVER 2A.
NOW, WE NEED TO SIMPLIFY,
WE CAN'T LEAVE OUR ANSWER
LIKE THIS,
SO LET'S FIGURE OUT
WHAT WE'VE GOT.
FIRST OF ALL, -(-1)
JUST BECOMES 1 + OR -,
SO WHAT DO WE HAVE
UNDER THE RADICAL?
WELL -1 SQUARED IS 1
AND 4 x 5 IS 20 x -3 IS -60.
SO THIS GUY IS NEGATIVE
BUT IT'S -(-60) SO + 60
ALL OVER 10,
WHICH IS 1 + OR - THE SQUARE
ROOT OF 61 ALL OVER 10,
SO THESE ARE MY TWO SOLUTIONS.
ALL RIGHT.
LET'S TRY ANOTHER ONE.
3R SQUARED + R = 10. OKAY.
SO A COMMON MISTAKE
THAT WE USUALLY SEE,
IS THAT STUDENTS
WILL IMMEDIATELY WRITE DOWN
THE A, B, AND C,
WITHOUT FIRST CHECKING
TO MAKE SURE THAT'S IT'S IN
STANDARD FORM.
THIS IS ALMOST IN STANDARD FORM
BUT WE NEED TO MOVE THIS GUY
TO THE LEFT ALSO,
SO IF I SUBTRACT 10
FROM BOTH SIDES,
I ACTUALLY GET 3R SQUARED PLUS R
- 10 = 0.
NOW, IT'S IN STANDARD FORM,
SO LET'S IDENTIFY, MY A IS 3,
MY B IS 1,
AND MY C IS -10.
PLUG THEM INTO THE FORMULA,
REMEMBER, THE FORMULA IS X
EQUALS
-B + OR - THE SQUARE ROOT
B SQUARED - 4AC ALL OVER 2A,
WHICH IS -1 + OR - THE SQUARE
ROOT OF 1 SQUARED - 4 x 3 x -10
ALL OVER 2 x 3,
WHICH IS -1 + OR -
THE SQUARE ROOT OF--
WE'VE GOT 1 AND THEN, 4 x 3
IS 12 x 10 IS 120,
BUT IT'S + SINCE YOU HAVE TWO
NEGATIVES SO + 120 ALL OVER 6.
SO IT'S -1 + OR - ROOT 121
ALL OVER 6.
THE SQUARE ROOT OF 121
IS ACTUALLY 11
SO THIS WORKS OUT NICE,
-1 + OR - 11 OVER 6.
WELL, SO IN THIS CASE, YOU
ACTUALLY WRITE OUT TWO ANSWERS,
-1 + 11 OVER 6, -1 - 11 OVER 6,
SO THIS ONE GETS THIS 10 OVER 6
WHICH IS 5/3
AND THIS ONE GIVES US -12 OVER 6
WHICH IS -2.
SO MY TWO ANSWERS ARE,
5/3 AND -2. OKAY.
ALL RIGHT, WE'RE GETTING
A LITTLE TOUGHER AS YOU CAN SEE
BUT STILL IT WON'T BECOME
ALL THAT BAD
AS LONG AS YOU PRACTICE
YOUR FORMULA.
LET'S TRY THE THIRD PROBLEM.
WE HAVE Y SQUARED + 4 = 6Y.
MOVE THE 6Y OVER,
NOTICE THAT ONE HAS TO GO
IN THE MIDDLE
BECAUSE THAT'S THE ORDER
OF THE TERMS,
SO LET'S WRITE DOWN OUR "A"
IS 1, OUR B IS -6, OUR C IS 4.
OKAY. SO WE HAVE OUR FORMULA.
X IS GOING TO EQUAL,
THIS TIME I'M NOT GOING TO WRITE
DOWN THE FORMULA AGAIN,
I GET -B + OR - THE SQUARE ROOT
OF B SQUARED - 4 x A x C
ALL OVER 2A, 2A.
THIS GIVES ME 6 + OR -
THE SQUARE ROOT OF 36 - 16
ALL OVER 2.
NOTICE THIS TIME, IT WAS A MINUS
BECAUSE WE ONLY HAD ONE NEGATIVE
THERE
WHICH IS 6 + OR -
THE SQUARE ROOT OF 20 OVER 2.
NOW, HERE WE CAN SIMPLIFY A BIT
BECAUSE THE SQUARE ROOT OF 20
CAN BE BROKEN DOWN.
REMEMBER, THE SQUARE ROOT OF 20
IS ACTUALLY 2 ROOT 5,
SO AFTER WE WRITE 6 + OR -
2 SQUARE ROOT OF 5 OVER 2.
NOW, WE'RE NEARLY DONE HERE.
WE CAN ACTUALLY FACTOR A 2
OUT OF THE TOP.
WHY WOULD WE DO THAT?
BECAUSE WE CAN CANCEL THE 2
AT THE BOTTOM,
WE END UP WITH Y = 3 + OR -
ROOT 5.
WHAT IF, WHEN I HAD THIS?
I JUST CANCEL THE 2s LIKE THIS,
NOTICE THAT WE END UP
WITH THE WRONG ANSWER. OKAY.
THE REASON IS BECAUSE YOU HAVE
TO CANCEL EVERYTHING ON THE TOP
EVEN THE 6,
SO THAT ACTUALLY WOULD GIVE US
THE RIGHT ANSWER.
EITHER WAY, I THINK, FACTORING
IS THE WAY TO PREVENT ERRORS.
OKAY. LET'S TRY ANOTHER ONE.
3/4 H SQUARED + 1/2H + 1/12 = 0.
THIS ONE'S NO FUN
BECAUSE WE HAVE FRACTIONS
ALL OVER THE PLACE,
BUT REMEMBER THE TRICK, RIGHT.
STEP TWO WAS IT?
YEAH, STEP TWO SAYS,
TO CLEAR ALL THE FRACTIONS
BY MULTIPLYING BY THE LCD.
IN THIS CASE, THE LCD IS 12,
SO I'M GOING TO MULTIPLY
EVERYTHING BY 12.
WHAT DO I END UP WITH?
WELL, LET ME JUST WRITE OUT 12
NEXT TO EVERYTHING
AND THIS I'M GOING TO CANCEL
SEPARATELY,
IT'S THE CLEAN WAY TO DO IT.
THIS BECOMES A 3 x 3H SQUARED
IS 9H SQUARED
AND THEN THE 2 AND THE 12
GIVE YOU A 6, SO 6H,
IT'S 1 AND THE 12, CANCEL
YOU GET 1
AND SO MY "A" IS 9, MY B IS 6,
AND MY C IS 1.
SO I PUT THEM
IN THE QUADRATIC FORMULA.
AGAIN, X = -6 + OR - THE SQUARE
ROOT OF 6 SQUARED - 4 x 9 x 1
ALL OVER 2 x 9,
WHICH IS -6 PLUS OR MINUS
THE SQUARE ROOT OF 36 - 36
ALL OVER 18.
BUT 36 - 36 IS 0.
SO I GET X = -6 + OR -
THE ROOT 0 OVER 18.
THE ROOT 0 IS JUST 0.
AGAIN, -6 + OR - 0
SO THAT'S JUST -6 OVER 18
WHICH IS -1/3.
NOTICE, I ONLY GOT ONE ANSWER.
THIS DOES HAPPEN SOMETIMES.
THIS JUST MEANS THAT WE HAVE
A REPEATED ROOT.
WE GOT TWO ANSWERS, THEY JUST
HAPPEN TO BE THE SAME ONE.
RIGHT?
BECAUSE REALLY,
WHAT WE HAVE IS -1/3 + OR - 0.
SO ACTUALLY,
THEY'RE THE SAME ANSWER.
OKAY, WE'RE GOING TO DO
ONE MORE EXAMPLE.
SO THE EXAMPLE IS GOING TO BE,
M x 5M - 2 = -3.
WHEN YOU HAVE SOMETHING
THAT LOOKS LIKE THIS,
YOU'VE ACTUALLY GOT DO
A LITTLE BIT OF WORK
TO PUT IT IN STANDARD FORM
FIRST.
THEY'RE ACTUALLY
MAKING YOUR LIFE DIFFICULT.
YOU NEED TO DISTRIBUTE THIS,
GET RID OF THOSE PARENTHESES.
WE GET 5N SQUARED - 2M = -3
AND THEN WE MOVE THIS GUY OVER.
WE GET 5M SQUARED - 2M + 3 = 0.
NOW, WE'RE READY TO GO, SO
WE GOT "A" = 5, B = -2, C = 3.
SO M IS GOING TO = -B + OR -
THE SQUARE ROOT OF B SQUARED
- 4 x A x C ALL OVER 2 x 5.
SO THAT BECOMES 2 + OR -
THE SQUARE ROOT OF 4
AND THEN 4 x 5 IS 20 x 3 IS 60
SO 4 - 60 ALL OVER 10
WHICH IS 2 + OR - THE SQUARE
ROOT OF -56 ALL OVER 10.
NOW, WE CAN STOP RIGHT HERE,
NOTICE THAT WE HAVE
A NEGATIVE UNDER A RADICAL.
THIS IS BEGINNING ALGEBRA
SO IN BEGINNING ALGEBRA,
WE CAN'T HAVE A NEGATIVE
UNDER THE RADICAL.
IN INTERMEDIATE ALGEBRA,
WE CAN HAVE A NEGATIVE
UNDER THE RADICAL
AND WE'LL LEARN
HOW TO DEAL WITH THAT.
BUT FOR BEGINNING ALGEBRA,
WE HAVE A NEGATIVE
UNDER THE RADICAL
AND WE CAN'T GO ANY FURTHER.
IN FACT, WHAT WE WRITE DOWN
IS NO REAL SOLUTION.
IN INTERMEDIATE ALGEBRA,
WE'LL LEARN HOW TO FIND
THE SOLUTIONS THAT ARE NOT REAL
BECAUSE THEY'RE STILL NOT REAL
IN THAT CLASS,
WE JUST KNOW HOW TO FIND THEM.
OKAY.
SO NO REAL SOLUTION.
MAKE SURE YOU CHECK CAREFULLY
THE SIGN OF THE RADICAND
TO MAKE SURE THAT THE SOLUTIONS
ACTUALLY ARE REAL,
IF NOT, YOU DON'T NEED TO WORRY
ABOUT THEM IN THIS CLASS.
OKAY, SO HOPEFULLY,
THIS HAS BEEN USEFUL.
THANK YOU VERY MUCH
FOR WATCHING.
