- WE WANT TO SOLVE 
EACH QUADRATIC EQUATION
BY FACTORING.
OUR FIRST EQUATION WE HAVE 
X SQUARED - 9 = 0.
HERE WE NEED TO RECOGNIZE THAT 
X SQUARED IS A PERFECT SQUARE
BECAUSE IT'S = TO X x X.
9 IS A PERFECT SQUARE 
BECAUSE IT'S = TO 3 x 3
AND WE HAVE A DIFFERENCE.
SO WE HAVE A DIFFERENCE 
OF SQUARES--
SO WE HAVE A DIFFERENCE 
OF SQUARES
AND IF IT'S HELPFUL 
WE CAN WRITE THIS
AS X SQUARED - 3 SQUARED 
EQUALS 0.
AND THEN IN THIS FORM 
WE CAN'T APPLY THE DIFFERENCE
OF SQUARE FORMULA,
WE'RE GOING TO HAVE 
1 FACTOR OF X + 3
AND 1 FACTOR OF X - 3.
AND NOW IF THIS PRODUCT 
IS = TO 0
THEN EITHER X + 3 = 0 
OR X - 3 = 0.
SO HERE IF WE SUBTRACT 3 
ON BOTH SIDES WE HAVE X = -3
OR HERE WE'D ADD 3 TO BOTH 
SIDES SO WE'D HAVE X = +3.
NOW, I SHOULD MENTION 
THAT SOMETIMES YOU'LL SEE
THESE SOLUTIONS WRITTEN  
AS X = + OR - 3.
THIS IS A SHORT WAY 
TO REPRESENT BOTH SOLUTIONS
OF +3 AND -3.
LOOKING AT THE SECOND EXAMPLE
WE SHOULD NOTICE THE FIRST 
TERM'S A PERFECT SQUARE
BECAUSE 9 X x 9 X 
IS = TO 81 X SQUARED.
THE SECOND TERM 
IS A PERFECT SQUARE
BECAUSE 8 x 8 IS = TO 64.
AND WE HAVE A DIFFERENCE,
WHICH MEANS THIS WILL FACTOR 
INTO 2 BINOMIAL FACTORS
AND IF IT'S HELPFUL 
WE CAN WRITE THIS
AS 9 X SQUARED - 8 SQUARED 
= 0.
SO THIS WILL FACTOR 
INTO 2 BINOMIAL FACTORS
WHERE 1 FACTOR WILL BE 9 X + 8
AND THE OTHER FACTOR WILL BE 
9 X - 8,
WHICH MEANS THIS PRODUCT 
WILL = 0
ONLY WHEN 9 X + 8 = 0 
OR WHEN 9 X - 8 IS = TO 0.
SO HERE WE WOULD SUBTRACT 8 
ON BOTH SIDES, DIVIDE BY 9
SO WE HAVE X = -8/9ths
AND HERE WE'LL ADD 8 
ON BOTH SIDES, DIVIDE BY 9
SO WE HAVE X = +8/9ths,
WHICH AGAIN COULD BE WRITTEN 
AS X = + OR - 8/9ths.
AND THEN AGAIN 
FOR THE LAST EXAMPLE
WE HAVE ANOTHER DIFFERENCE 
OF SQUARES
BECAUSE 25 IS A PERFECT SQUARE
FOR A--SQUARED 
AS A PERFECT SQUARE
AND WE HAVE A DIFFERENCE.
AND SINCE 5 x 5 IS = TO 25
AND 2 X x 2 X 
IS = TO 4 X SQUARED
1 FACTOR WOULD BE 5 + 2 X
AND 1 FACTOR WOULD BE 5 - 2 X.
SO THIS PRODUCT WILL BE 0 
WHEN 5 + 2 X = 0
OR WHEN 5 - 2 X = 0.
SO HERE WE WOULD SUBTRACT 5 
ON BOTH SIDES, DIVIDE BY 2
SO WE HAVE X = -5 HALVES
OR HERE WE WOULD SUBTRACT 5 
ON BOTH SIDES
AND THEN DIVIDE BY -2.
SO HERE WE HAVE X = +5 HALVES
OR IF WE WANT WE CAN SAY 
X = +OR - 5 HALVES.
OKAY, HOPE THIS HELPS.
