- 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.