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- WE WANT TO SOLVE THE GIVEN EQUATIONS BY FACTORING.
REMEMBER, THE FIRST STEP IN FACTORING
IS TO FACTOR OUT THE GREATEST COMMON FACTOR.
SO FOR 12X SQUARED + 18X = 0,
WE WANT TO FACTOR OUT THE GREATEST COMMON FACTOR
OF THESE TWO TERMS WHICH WOULD BE 6X.
BUT JUST IN CASE WE HAVE A HARD TIME
DETERMINING THE GREATEST COMMON FACTOR,
IT MAY BE HELPFUL TO EXPAND THESE INTO PRIME FACTOR FORM.
SO TO FIND THE PRIME FACTORIZATION OF 12,
WE'LL MAKE A FACTOR TREE.
12 IS EQUAL TO 4 x 3 AND 4 IS EQUAL TO 2 x 2.
SO AGAIN IF IT'S HELPFUL,
WE COULD WRITE 12X SQUARED AS 2 x 2 x 3
AND THEN X SQUARED IS EQUAL TO X x X,
PLUS FOR THE PRIME FACTORIZATION OF 18,
WE'D HAVE 2 x 9, 2 IS PRIME AND 9 IS EQUAL TO 3 x 3,
SO FOR 18X, WE'D HAVE 2 x 3 x 3 x X.
NOW AGAIN, WE DON'T HAVE TO SHOW THIS WORK.
BUT IF WE AREN'T ABLE TO IDENTIFY
THE GREATEST COMMON FACTOR BY INSPECTION,
THIS IS VERY HELPFUL
BECAUSE NOW WE CAN SEE ALL THE COMMON FACTORS.
THESE TWO TERMS HAVE A COMMON FACTOR OF 2, 3 AND X,
SO 6X WOULD BE THE GREATEST COMMON FACTOR
WHICH IS WHAT WE NEED TO FACTOR OUT.
SO IF WE FACTOR OUT 6X,
WE CAN SEE WE'RE GOING TO BE LEFT WITH 2X + 3.
AND SINCE THIS PRODUCT IS EQUAL TO 0,
WE CAN USE THE 0 PRODUCT PROPERTY
TO NOW SOLVE THIS EQUATION.
IF THIS PRODUCT IS EQUAL TO 0,
EITHER THE FIRST FACTOR OF 6X MUST EQUAL 0,
OR THE SECOND FACTOR OF 2X + 3 MUST EQUAL 0.
SO TO SOLVE THIS FIRST EQUATION FOR X,
WE WOULD DIVIDE BOTH SIDES BY 6,
BUT 0 DIVIDED BY 6 IS STILL 0.
SO WE HAVE X = 0
OR TO SOLVE THIS EQUATION FOR X,
WE WOULD START BY SUBTRACTING 3 ON BOTH SIDES.
SO WE'D HAVE 2X = NEGATIVE 3
AND NOW DIVIDE BOTH SIDES BY 2,
SO WE HAVE X = A NEGATIVE 3 HALVES.
THESE WOULD BE THE TWO SOLUTIONS
TO THE GIVEN EQUATION.
NOW LOOKING AT OUR SECOND EXAMPLE,
IF WE WANT TO SOLVE THIS EQUATION BY FACTORING,
WE FIRST NEED TO SET THE EQUATION EQUAL TO 0.
SO THE FIRST STEP HERE WILL BE TO SUBTRACT 20X
ON BOTH SIDES OF THE EQUATION.
AND THESE ARE NOT LIKE TERMS,
SO WE'LL HAVE 4X SQUARED - 20X = 0.
AND NOW, WE WANT TO IDENTIFY THE GREATEST COMMON FACTOR
OF 4X SQUARED AND 20X WHICH WOULD BE 4X.
BUT JUST IN CASE WE'RE NOT ABLE TO IDENTIFY
THE GREATEST COMMON FACTOR,
IT MAY BE HELPFUL TO WRITE 4X SQUARED AS 2 x 2 x X x X,
NUMBER 20X,
SINCE 20 IS 4 x 5, WE WOULD HAVE 2 x 2 x 5 x X.
AND AGAIN IN THIS FORM, EVEN THOUGH IT'S NOT NECCE--
AND THEN IN THIS FORM, EVEN THOUGH IT'S NOT NECESSARILY,
WE CAN ACTUALLY SEE THE COMMON FACTORS.
BOTH OF THESE TERMS HAVE TWO FACTORS OF 2 IN COMMON
AS WELL AS ONE FACTOR OF X WHICH AGAIN VERIFIES
THE GREATEST COMMON FACTOR WOULD BE 4X.
SO IF WE FACTOR OUT 4X,
WE'RE GOING TO BE LEFT WITH X - 5 = 0.
AND AGAIN IF THIS PRODUCT IS EQUAL TO 0,
THEN EITHER THE FIRST FACTOR OF 4X MUST = 0
OR THE SECOND FACTOR OF X - 5 MUST EQUAL 0.
AND NOW, WE'LL SOLVE THESE TWO EQUATIONS FOR X
TO FIND OUR SOLUTIONS.
SO HERE WE DIVIDE BOTH SIDES BY 4.
THIS WOULD GIVE US X = 0.
AND FOR THE SECOND EQUATION, WE WOULD ADD 5 TO BOTH SIDES.
THIS WOULD GIVE US X = POSITIVE 5.
SO OUR TWO SOLUTIONS ARE X = 0 OR X = 5.
OKAY. I HOPE YOU FOUND THIS HELPFUL.