- WE WANT TO SOLVE 
THE GIVEN QUADRATIC EQUATIONS
BY FACTORING.
NOTICE ALL THE EQUATIONS 
ARE IN THE FORM
AX SQUARED + BX + C = 0 
WHEN A = 1
MEANING THE FIRST TERM 
IS X SQUARED.
SO LOOKING AT OUR FIRST EXAMPLE,
REMEMBER 
THE FIRST STEP IN FACTORING
IS TO LOOK 
FOR A GREATEST COMMON FACTOR.
THE ONLY COMMON FACTOR THESE 
TERMS SHARE IS 1.
SO IF THIS DOES FACTOR 
IT'LL FACTOR INTO TWO BINOMIALS
WHERE THE FIRST TERMS COME 
FROM THE FACTORS OF X SQUARED
WHICH WOULD BE X AND X.
AND THEN THE SECOND TERMS 
OF THE BINOMIAL FACTORS
WOULD COME FROM THE FACTORS 
OF 8 THAT ADD TO 6
OR THE FACTORS OF C 
THAT ADD TO B.
AND SINCE 2 x 4 IS = 8
AND 2 + 4 = 6; 
2 AND 4 ARE THE WINNING FACTORS.
AND SINCE THE 2 AND 4 
ARE BOTH POSITIVE
WE WOULD HAVE X + 2 x X + 4.
NOW THAT WE HAVE THE TRINOMIAL 
FACTORED
WE CAN USE THE ZERO PRODUCT 
PROPERTY TO SOLVE THIS EQUATION.
IF THIS PRODUCT IS EQUAL TO 0
THEN X + 2 MUST = 0 
OR THE FACTOR X + 4 MUST = 0.
AND NOW WE SOLVED 
THESE TWO EQUATIONS FOR X.
SO WE'LL SUBTRACT 2 
ON BOTH SIDES HERE.
WE HAVE X = -2 OR SUBTRACT 4 ON 
BOTH SIDES HERE, WE HAVE X = -4.
THESE WOULD BE OUR TWO SOLUTIONS 
TO THE QUADRATIC EQUATION.
NOW LOOKING 
AT THE SECOND EXAMPLE,
AGAIN THE ONLY COMMON FACTOR 
BETWEEN THESE THREE TERMS IS 1.
SO IF THIS IS GOING TO FACTOR 
IT'LL FACTOR INTO TWO BINOMIALS.
THE FIRST TERMS COME 
FROM THE FACTORS OF X SQUARED,
THAT WOULD BE X AND X.
AND THE SECOND TERMS WILL BE THE 
FACTORS OF -50 THAT ADD TO 23.
AND SINCE -2 x 25 = -50
AND -2 + 25 DOES = 23
THE TWO FACTORS WE NEED 
ARE -2 AND 25.
SO ONE FACTOR WILL BE X - 2.
THE OTHER FACTOR WILL BE X + 25.
AND THAT'LL SOLVE 
SINCE THIS PRODUCT = 0
EITHER X - 2 MUST = 0 
OR X + 25 MUST = 0.
SO HERE WE'LL ADD 2 
TO BOTH SIDES
SO IF X = 2 OR SUBTRACT 25 
ON BOTH SIDES WE HAVE X = -25.
NEXT EXAMPLE, 
THE ONLY COMMON FACTOR IS 1
SO AGAIN IF THIS DOES FACTOR, 
IT'LL FACTOR INTO TWO BINOMIALS.
THE FIRST TERM OF EACH FACTOR
WILL COME FROM THE FACTORS OF X 
SQUARED, THAT'S X AND X.
THE SECOND TERMS WILL COME
FROM THE FACTORS OF 15 
THAT ADD TO -8.
AND SINCE THE SUM OF THE FACTORS 
MUST BE NEGATIVE,
BUT THE PRODUCT MUST BE POSITIVE
WE'LL HAVE TO USE TWO 
NEGATIVE FACTORS OF 15.
FOR EXAMPLE, -3 x -5 IS = 15, 
BUT -3 + -5 DOES GIVE US -8,
THE COEFFICIENT 
IN THE MIDDLE TERM.
SO OUR TWO FACTORS 
WILL BE X - 3 AND X - 5.
AND NOW TO SOLVE THIS EQUATION,
SINCE THIS PRODUCT IS EQUAL TO 0
EITHER X - 3 MUST = 0 
OR X - 5 MUST = 0.
SO HERE WE'LL ADD 3 
TO BOTH SIDES OF THE EQUATION.
WE HAVE X = 3 OR ADD 5 TO BOTH 
SIDES AND WE HAVE X = 5.
LET'S GO AHEAD AND TAKE A LOOK 
AT ONE MORE EXAMPLE.
THERE ARE NO COMMON FACTORS
SO THIS WILL FACTOR 
INTO TWO BINOMIALS.
THE FIRST TERMS WILL COME FROM 
THE FACTORS OF X SQUARED.
SO WE HAVE X AND X 
AND NOW WE WANT
THE FACTORS OF -24 
THAT ADD TO -5.
WELL, LET'S SEE -3 x 8 
WOULD BE -24,
BUT THE SUM WOULD BE 5.
WE WANT A -5 
SO IF WE USE A -8 AND A 3
THIS WOULD GIVE US -24 
AND THE SUM WOULD BE -5.
SO THESE ARE THE TWO
WINNING FACTORS.
SO WE HAVE X - 8 x X + 3.
THEREFORE, THE SOLUTIONS OCCUR 
WHEN X - 8 = 0
OR X + 3 = 0, 
ADD 8 TO BOTH SIDES.
HERE WE HAVE X = 8, 
SUBTRACT 3 ON BOTH SIDES
AND HERE WE HAVE X = -3.
OKAY, I HOPE YOU FOUND 
THESE EXAMPLES HELPFUL.
