- WE WANT TO SOLVE
THE GIVEN QUADRATIC EQUATION
USING THE QUADRATIC FORMULA.
SO THE FIRST THING WE SHOULD
RECOGNIZE ABOUT THIS EXAMPLE
IS THAT THE EQUATION
IS NOT IN THE CORRECT FORM
IN ORDER TO IDENTIFY
THE VALUES OF "A", B, AND C.
WE NEED THE EQUATION TO BE
IN THIS FORM
WHERE THE FORM "AX" SQUARED
+ BX + C = 0.
SO FOR THE FIRST STEP
WE WANT TO SET THIS EQUATION
EQUAL TO 0.
SO WE'LL ADD 12X
TO BOTH SIDES,
AND THEN SUBTRACT 3
ON BOTH SIDES,
WHICH WOULD GIVE US
THE EQUATION
9X SQUARED + 12X - 3 = 0.
SO IN THIS FORM
WE CAN NOW RECOGNIZE
THAT "A" = 9, B = 12,
AND C = -3.
SO WE'LL USE THESE VALUES
TO PERFORM SUBSTITUTION
INTO THE QUADRATIC FORMULA.
SO WE'LL HAVE X = -B OR -12
+ OR - SQUARE ROOT
OF B SQUARED - 4AC,
WHICH WILL BE 12 SQUARED, - 4
x 9 x -3 DIVIDED BY 2 x "A"
OR IN THIS CASE 2 x 9.
NOTICE HOW FOR THIS FIRST STEP
WE JUST PERFORMED
THE SUBSTITUTION.
NOW WE'LL BEGIN TO SIMPLIFY.
SO NOW WE'LL HAVE X = -12
+ OR - THE SQUARE ROOT.
12 SQUARED IS 144.
THEN WE HAVE - 4 x 9 x -3.
THAT WILL BE - -108
OR + 108 DIVIDED BY 2 x 9,
WHICH IS 18.
SO NOW WE HAVE X = -12
+ OR - SQUARE ROOT
OF 144 + 108 = 252.
AND NOW FOR THE NEXT STEP
WE WANT TO SIMPLIFY
THE SQUARE ROOT OF 252
BY IDENTIFYING ALL OF THE
PERFECT SQUARE FACTORS OF 252.
TO DO THIS WE'RE GOING TO FIND
THE PRIME FACTORIZATION
OF 252.
SO 252 = 2 x 126 = 2 x 63
= 9 x 7, AND 9 = 3 x 3.
SO THE PRIME FACTORIZATION
WOULD BE 2 x 2 x 3 x 3 x 7.
EVERY TIME WE HAVE
TWO EQUAL FACTORS
THE SQUARE ROOT WILL SIMPLIFY.
THIS IS A PERFECT SQUARE
FACTOR, AND SO IS THIS.
THE SQUARE ROOT OF 2 x 2 IS 2,
AND THE SQUARE ROOT OF 3 x 3
IS 3,
AND WE STILL HAVE
SQUARE ROOT 7.
SO THIS SIMPLIFIES
TO 6 SQUARE ROOT 7.
NOW WE HAVE X = -12 + OR - 6
SQUARE ROOT 7/18.
NOW THERE'S A COUPLE OF WAYS
OF SIMPLIFYING THIS,
AND WE'LL SHOW BOTH.
BUT FOR THE FIRST METHOD
WE'RE GOING TO BREAK THIS UP
INTO TWO SEPARATE FRACTIONS,
SINCE WE'RE DIVIDING
BY A MONOMIAL.
THIS IS THE SAME AS -12/18
+ OR - 6 SQUARE ROOT 7/18.
IN THIS FORM WE'LL SIMPLIFY
EACH FRACTION SEPARATELY.
THE COMMON FACTOR OF 6
IN 12 AND 18,
THERE ARE 3 SIXES IN 18
AND 2 SIXES IN 12.
AGAIN WE HAVE A COMMON FACTOR
OF 6.
THERE ARE 3 SIXES IN 18
AND 1 SIX IN SIX.
SO OUR SOLUTIONS ARE X = -2/3
+ OR - SQUARE ROOT 7/3.
WE HAVE TWO SOLUTIONS HERE.
BOTH OF THESE ARE
IN SIMPLIFIED FORM.
ONE SOLUTION
IS -2/3 + SQUARE ROOT 7/3.
THE OTHER SOLUTION
IS -2/3 - SQUARE ROOT 7/3.
BUT LET'S GO AHEAD
AND SHOW THE SECOND METHOD
FOR SIMPLIFYING OUR SOLUTIONS.
SO LET'S GO AHEAD
AND CONSIDER THIS FORM HERE.
WE NEED TO BE CAREFUL,
BECAUSE IT WOULD BE A MISTAKE
TO JUST SIMPLIFY THE 12
AND THE 18 HERE.
WE CANNOT SIMPLIFY ACROSS
ADDITION OR SUBTRACTION.
SO IF WE LOOK
AT THE NUMERATOR,
NOTICE THAT 12 AND 6
SHARE A COMMON FACTOR OF 6.
SO WE COULD WRITE THIS
AS X = 6 x THE QUANTITY -2
+ OR - SQUARE ROOT 7.
AND THEN WE KNOW
THAT 18 = 6 x 3,
AND BECAUSE OUR FACTORS
ARE BEING MULTIPLIED
WE CAN SIMPLIFY THIS.
6/6 SIMPLIFIES TO 1.
SO OUR SOLUTION CAN
ALSO BE EXPRESSED
AS X = -2 + OR - SQUARE ROOT
7/3.
SO BOTH OF THESE SOLUTIONS
ARE EQUIVALENT,
EVEN THOUGH THEY ARE
IN A SLIGHTLY DIFFERENT FORM.
OKAY. I HOPE YOU FOUND
THIS EXAMPLE HELPFUL.
