- 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.
