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X
- 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.