- WE WANT TO FACTOR AND SOLVE 
THE GIVEN QUADRATIC EQUATIONS.
THE MOST IMPORTANT THING 
TO REMEMBER
WHEN SOLVING A QUADRATIC 
EQUATION BY FACTORING
IS TO NOT FORGET THE FIRST STEP 
IN FACTORING,
WHICH IS THE FACTOR OUT 
THE GREATEST COMMON FACTOR.
IN OUR FIRST EXAMPLE WE HAVE 
3X SQUARED + 15X + 18 = 0.
SO IT'S IMPORTANT TO RECOGNIZE 
IN THIS PROBLEM
THAT 3 IS THE GREATEST COMMON 
FACTOR OF THESE THREE TERMS,
SO THE FIRST STEP 
IS TO FACTOR THE 3 OUT.
AND BY DOING THIS,
IT'S GOING TO MAKE IT A MUCH 
EASIER FACTORING PROBLEM.
IF WE FACTOR OUT 3, WE'D BE LEFT 
WITH X SQUARED + 5X + 6 = 0.
AND THEN WE CAN FACTOR 
THE TRINOMIAL FACTOR
AS WE NORMALLY WOULD,
THE LEADING COEFFICIENT OF 1.
SO WE'D HAVE THIS FACTOR OF 3, 
AND THEN IF THIS DOES FACTOR,
IT WILL FACTOR TO 
TWO BINOMIAL FACTORS.
AND BECAUSE THE LEADING 
COEFFICIENT IS 1,
IT MAKES IT A LOT EASIER 
TO FACTOR.
THE FACTORS OF X SQUARED 
GO IN THE FIRST POSITIONS.
AND NOW WE WANT THE FACTORS 
OF 6 THAT ADD TO 5.
WELL, 3 x 2 = 6, AND 3 + 2 = 5.
THEREFORE, ONE BINOMIAL FACTOR 
IS GOING TO X + 3,
AND ONE OF THE FACTORS 
WILL BE X + 2.
SO NOW THAT WE HAVE 
IT IN FACTORED FORM,
AND THIS PRODUCT = 0, 
THE FACTOR OF X + 3 MUST = 0,
OR THE FACTOR OF X + 2 MUST = 0.
SO NOW IF WE SOLVE THIS EQUATION 
FOR X, WE WOULD HAVE X = -3,
BY SUBTRACTING 3 ON BOTH SIDES.
AND HERE WE WOULD SUBTRACT 2 
ON BOTH SIDES,
FOR A SOLUTION OF X = -2.
SO NOTICE HOW IF WE MISSED 
THIS COMMON FACTOR OF 3,
WE WOULD MAKE FACTORING THIS 
MUCH MORE DIFFICULT.
LOOKING AT OUR SECOND EXAMPLE,
WE HAVE 8X SQUARED 
- 72X + 162 = 0.
AGAIN, THE FIRST STEP 
IS TO LOOK FOR COMMON FACTORS.
IT WOULD BE NICE IF 8 
WAS A COMMON FACTOR,
BUT 8 DOES NOT DIVIDE EVENLY 
INTO 162.
SO I BELIEVE OUR ONLY COMMON 
FACTOR HERE IS 2.
LET'S START BY FACTORING OUT 
A 2, AND SEE WHAT WE HAVE LEFT.
IF WE FACTOR 2 OUT, WE'D BE LEFT 
WITH 4X SQUARED - 36X + 81 = 0.
NOW, EVEN THOUGH 
THIS TRINOMIAL FACTOR HERE
DOES NOT HAVE A LEADING 
COEFFICIENT OF 1,
NOTICE HOW THE FIRST TERM 
IS A PERFECT SQUARE,
AND SO IS THE THIRD TERM.
SO LET'S TRY FACTORING THIS 
AS A PERFECT SQUARE TRINOMIAL.
SO WE'LL STILL HAVE THIS EXTRA 
FACTOR OF 2,
AND THEN WE'LL HAVE 
TWO BINOMIAL FACTORS.
AND BECAUSE THE FIRST TERM 
IS A PERFECT SQUARE,
WE'LL USE THE FACTORS OF 2X 
AND 2X FOR 4X SQUARED.
AND THEN SINCE 
81 IS A PERFECT SQUARE,
WE USE THE FACTORS OF 9 AND 9.
AND THEN WE USE A SIGN 
ON THE MIDDLE TERM,
SO WE'LL MAKE BOTH 
OF THESE MINUS.
NOW, TO MAKE SURE THIS 
ACTUALLY WORKS,
WE'LL CHECK THE SUM 
OF THE INNER AND OUTER PRODUCT
TO MAKE SURE THAT WE HAVE -36X.
THE INNER PRODUCT IS -18X.
THE OUTER PRODUCT IS -18X AS 
WELL, WHICH DOES GIVE US -36X.
SO THIS IS FACTORED CORRECTLY.
SO NOTICE HOW WHAT LOOKED LIKE
A PRETTY CHALLENGING 
FACTORING PROBLEM
BECAME MUCH MORE MANAGEABLE,
ONCE WE FACTORED OUT THE 
GREATEST COMMON FACTOR OF 2.
AND WE CAN REWRITE 
THIS ONE MORE TIME.
SINCE WE DO HAVE TWO FACTORS 
OF 2X - 9,
WE COULD WRITE THIS AS 2 
x THE QUANTITY 2X - 9 SQUARED
RAISED TO THE SECOND = 0.
NOW, THIS IS ONLY GOING TO = 0 
WHEN THE FACTOR OF 2X - 9 = 0.
SO WE WOULD ADD 9 TO BOTH SIDES.
SO WE'D HAVE 2X = 9. 
DIVIDE BOTH SIDES BY 2.
SO WE HAVE ONE SOLUTION. 
IT'S X = 9/2.
AND WE MENTIONED BEFORE THAT 
SINCE WE DID HAVE TWO FACTORS
OF 2X - 9,
WE CAN SAY THAT THE SOLUTION 
HAS A MULTIPLICITY OF 2,
OR THAT IT'S A DOUBLE 0 
OR DOUBLE ROOT.
WE'LL TAKE A LOOK AT SOME MORE 
EXAMPLES IN THE NEXT VIDEO.
