- WE WANT TO SOLVE 
THE QUADRATIC EQUATIONS
BY FACTORING.
REMEMBER THE FIRST STEP 
IN ANY FACTORING PROBLEM
IS TO FACTOR OUT 
THE GREATEST COMMON FACTOR.
LOOKING AT THE FIRST EXAMPLE 
WE HAVE 5 X SQUARED - 45 = 0.
WE NEED TO RECOGNIZE HERE
THAT 5 IS THE GREATEST COMMON 
FACTOR OF THESE 2 TERMS.
SO WE CAN REWRITE THIS 
AS 5 X THE QUANTITY X SQUARED
AND THERE ARE 9-5'S IN 45 
SO WE HAVE X SQUARED - 9 = 0.
NOTICE ONCE WE FACTOR OUT 
THE 5
THE REMAINING BINOMIAL FACTOR 
IS A DIFFERENCE OF SQUARES.
X SQUARED IS A PERFECT SQUARE, 
9'S A PERFECT SQUARE
AND WE HAVE A DIFFERENCE.
SO THIS IS GOING TO FACTOR 
INTO 2 BINOMIAL FACTORS
WHERE 1 FACTOR IS GOING TO BE 
X + 3
AND 1 FACTOR WILL BE X - 3.
AGAIN, THIS COMES FROM THE 
DIFFERENCE OF SQUARES FORMULA.
X x X IS = TO X SQUARED, 
3 x 3 IS = TO 9
AND WE HAVE A DIFFERENCE 
OF SQUARES,
ONE'S A SUM 
AND ONE'S A DIFFERENCE.
NOW, THIS PRODUCT 
IS ONLY GOING TO = 0
WHEN X + 3 = 0 
OR WHEN X - 3 = 0.
THERE'S NO VARIABLE HERE 
WITH THE 5
SO THIS IS NOT GOING 
TO GIVE US A SOLUTION.
SO HERE WE WOULD SUBTRACT 3 
ON BOTH SIDES,
WE HAVE X = -3 OR ADD 3 
TO BOTH SIDES WE HAVE X = +3.
AND THIS CAN BE WRITTEN 
AS X = + OR - 3 IF WE WANT.
NOW, LOOKING 
AT OUR SECOND EXAMPLE,
NOTICE THAT A IS NOT = TO 1
WHICH USUALLY MAKES 
THE FACTORING PROBLEM
MORE DIFFICULT.
HOWEVER, NOTICE THAT 3 
IS THE GREATEST COMMON FACTOR
AND THEREFOR CAN BE 
FACTORED OUT.
SO THE FIRST STEP HERE 
IS TO FACTOR OUT THE 3
LEAVING US 
WITH X SQUARED - 2 X
AND THERE ARE 15-3'S IN 45.
SO WE HAVE - 15 = 0.
AND THEN NOTICE 
THE TRINOMIAL FACTOR
HAS A LEADING COEFFICIENT 
OF 1.
SO LET'S SEE IF WE CAN FACTOR 
THIS INTO 2 BINOMIAL FACTORS.
WE WOULD HAVE 
3-2 BINOMIAL FACTORS = 0.
WELL, THE FACTORS OF X SQUARED 
ARE X AND X
AND NOW WE WANT THE FACTORS 
OF -15 THAT ADD TO -2.
AND -5 X +3 IS = TO -15 
AND -5 + 3 IS = TO -2.
SO THE 2 FACTORS THAT SATISFY 
THESE CONDITIONS
ARE -5 AND +3.
SO ONE FACTOR IS GOING TO BE 
X - 5,
THE OTHER FACTOR 
IS GOING TO BE X + 3,
WHICH MEANS THIS PRODUCT 
WILL = 0
WHEN X - 5 = 0 
OR WHEN X + 3 = 0.
SO HERE WE'LL ADD 5 
TO BOTH SIDES, X = 5
OR SUBTRACT 3 ON BOTH SIDES, 
X = -3.
SO THE MOST IMPORTANT THING 
TO REMEMBER ABOUT FACTORING
IS THE FIRST STEP
IS TO FACTOR OUT 
THE GREATEST COMMON FACTOR.
IF WE DON'T DO THIS
IT'S GOING TO MAKE FACTORING 
THESE PROBLEMS
SEEM MUCH MORE DIFFICULT 
THEN IT HAS TO BE.
