- NOW, WE'LL LOOK AT SOME 
EXAMPLES OF FACTORING
AND SOLVING QUADRATIC EQUATIONS
IN THE FORM 
X SQUARED + BX + C = 0.
NOTICE HOW THE LEADING 
COEFFICIENT IS EQUAL TO 1.
OUR FIRST EQUATION, WE HAVE 
X SQUARED + 4X - 21 = 0.
SO IF THIS TRINOMIAL FACTORS,
IT WILL FACTOR INTO TWO BINOMIAL 
FACTORS
WHERE THE TERMS 
IN THE FIRST POSITIONS
WILL COME FROM THE FACTORS OF X 
SQUARED WHICH WOULD BE X AND X,
AND THE TERMS 
IN THE SECOND POSITIONS
WILL BE THE FACTORS OF C 
THAT ADD TO B,
OR IN THIS CASE THE FACTORS 
OF NEGATIVE 21
THAT ADD TO POSITIVE 4.
LET'S LIST ALL THE FACTORS 
THAT HAVE NEGATIVE 21.
WE COULD HAVE NEGATIVE 1 x 21
OR NEGATIVE 21 x 1 OR NEGATIVE 3 
x 7 OR NEGATIVE 7 x 3.
WE WANT THE TWO FACTORS THAT 
HAVE A SUM OF POSITIVE 4,
WHICH MEANS THE WINNING FACTORS
WILL BE NEGATIVE 3 
AND POSITIVE 7.
THEREFORE ONE OF THE BINOMIAL 
FACTORS WILL BE X - 3
AND THE OTHER FACTOR 
WILL BE X + 7.
BECAUSE THE 3 IS NEGATIVE, 
WE HAVE -3.
BECAUSE THE 7 IS POSITIVE, 
WE HAVE 7.
OF COURSE IF WE PUT THE 7 HERE 
AND THE -3 HERE,
IT'S NOT GOING TO CHANGE 
OUR SOLUTIONS.
SO NOW, WE HAVE THE TRINOMIAL 
IN FACTORED FORM
WHERE THIS PRODUCT 
IS EQUAL TO ZERO,
WE CAN USE THE ZERO PRODUCT 
PROPERTY TO SOLVE THIS EQUATION.
AGAIN IF THIS PRODUCT 
IS EQUAL TO 0,
EITHER THE FIRST FACTOR 
OF X - 3 MUST EQUAL 0
OR THE SECOND FACTOR 
OF X + 7 MUST EQUAL 0.
NOW, WE'LL SOLVE 
EACH OF THESE EQUATIONS FOR X.
SO HERE WE WOULD ADD 3 
TO BOTH SIDES.
SO WE'D HAVE X = 3, OR HERE WE 
WOULD SUBTRACT 7 ON BOTH SIDES,
SO WE'D HAVE X = NEGATIVE 7.
THESE ARE THE TWO SOLUTIONS 
TO OUR QUADRATIC EQUATION.
LET'S TAKE A LOOK AT A SECOND 
EXAMPLE.
AGAIN IF THIS DOES FACTOR, IT'LL 
FACTOR INTO TWO BINOMIAL FACTORS
WHERE THE TERMS IN THE FIRST 
POSITIONS
WILL BE THE FACTORS OF X SQUARED 
WHICH WILL BE X AND X.
AND THEN THE TERMS 
IN THE SECOND POSITIONS
WILL BE THE FACTORS OF NEGATIVE 
27 THAT ADD TO NEGATIVE 6.
LET'S GO AHEAD AND LIST 
THE FACTORS OF NEGATIVE 27.
WE COULD HAVE NEGATIVE 3 x 9, 
NEGATIVE 9 x 3, NEGATIVE 1 x 27,
AND NEGATIVE 27 x 1.
OF COURSE IF WE CAN VISUALIZE 
THESE IN OUR MIND,
WE DON'T HAVE TO LIST THEM.
BUT NOW WE WANT THE TWO FACTORS 
THAT HAVE A SUM OF NEGATIVE 6
WHICH WOULD BE NEGATIVE 9 
AND POSITIVE 3
WHICH MEANS ONE FACTOR 
WILL BE X - 9
AND ONE FACTOR WILL BE X + 3.
AND NOW THAT 
IT'S IN FACTORED FORM
AND WE HAVE OUR PRODUCT 
EQUAL TO 0,
EITHER THE FIRST FACTOR OF X - 9
MUST EQUAL 0 OR THE FACTOR 
OF X + 3 MUST EQUAL 0.
AND NOW, WE'LL SOLVE 
THESE TWO EQUATIONS FOR X
TO DETERMINE OUR SOLUTIONS.
SO HERE WE WOULD ADD 9 
TO BOTH SIDES.
WE'D HAVE X = 9 OR HERE WE WOULD 
SUBTRACT 3 ON BOTH SIDES,
SO WE'D HAVE X = NEGATIVE 3.
AGAIN, 
THESE ARE THE TWO SOLUTIONS
TO OUR QUADRATIC EQUATION.
OKAY, WE'LL LOOK AT SOME MORE 
EXAMPLES IN THE NEXT VIDEO.
