- WE WANT TO FACTOR AND SOLVE
QUADRATIC EQUATIONS
IN THE FORM
A SQUARED - B SQUARED = 0
WHICH MEANS WHEN WE HAVE
A DIFFERENCE OF SQUARES.
LET'S REVIEW
HOW WE FACTOR THIS FIRST.
IF WE HAVE A BINOMIAL IN THE
FORM OF A SQUARED - B SQUARED,
THIS WILL FACTOR
INTO TWO BINOMIAL FACTORS
WHERE ONE BINOMIAL FACTOR
WILL BE "A" + B
AND ONE FACTOR
WILL BE "A" - B.
SO IF WE TAKE A LOOK
AT OUR FIST EXAMPLE,
WE HAVE 48X SQUARED - 75.
NOTICE 48X SQUARED AND 75
ARE NOT PERFECT SQUARES
AND THEREFORE WE MAY THINK
THIS WON'T FACTOR,
BUT WE'RE FORGETTING
THE FIRST STEP IN FACTORING.
THE FIRST STEP IN FACTORING
IS TO FACTOR OUT
THE GREATEST COMMON FACTOR,
AND BOTH OF THESE TERMS
DO HAVE A COMMON FACTOR OF 3.
48X SQUARED CAN BE WRITTEN
AS 3 x 16X SQUARED
AND 75 CAN BE WRITTEN
AS 3 x 25.
SO THE FIRST STEP
IN THIS PROBLEM
IS TO FACTOR OUT THE GREATEST
COMMON FACTOR OF 3.
IF WE DO THIS, WE'LL HAVE 3 x
THE QUANTITY 16X SQUARED - 25
IS EQUAL TO 0.
AND NOW,
NOTICE OUR BINOMIAL FACTOR
IS A DIFFERENCE OF SQUARES.
16X SQUARED
IS A PERFECT SQUARE,
BECAUSE IT'S EQUAL TO 4X
RAISED TO THE SECOND POWER.
25 IS A PERFECT SCORE, BECAUSE
IT'S EQUAL TO 5 SQUARED.
WRITTEN IN THIS FORM,
WE CAN SEE "A" IS EQUAL TO 4X
AND B IS EQUAL TO 5.
SO NOW IF WE CAN FACTOR THIS
AGAIN,
WE WOULD HAVE 3 x 2
BINOMIAL FACTORS = 0
WHERE ONE BINOMIAL FACTOR
WOULD BE 4X + 5
AND ONE BINOMIAL FACTOR
WOULD BE 4X - 5.
NOTICE HOW WE HAVE A PRODUCT
NOW THAT'S EQUAL TO 0.
THEREFORE BY USING
THE 0 PRODUCT PROPERTY,
THE FACTOR OF 4X + 5
MUST EQUAL 0
OR THE FACTOR OF 4X - 5
MUST EQUAL 0.
NOTICE THE FACTOR OF 3
DOES NOT CONTAIN A VARIABLE
AND THEREFORE WILL NOT GIVE US
A SOLUTION TO THIS EQUATION.
AND NOW, THE LAST STEP IS
TO SOLVE THESE TWO EQUATIONS
FOR X.
SO FOR THIS FIRST EQUATION,
WE'LL START BY SUBTRACTING 5
ON BOTH SIDES.
THIS WILL GIVE US
4X = NEGATIVE 5,
DIVIDE BOTH SIDES BY 4,
SO WE HAVE X = NEGATIVE 5/4.
AND NOW, WE'LL SOLVE
THE SECOND EQUATION FOR X.
SO WE'LL ADD 5 TO BOTH SIDES,
SO WE HAVE 4X = 5,
DIVIDE BOTH SIDES BY 4,
AND WE HAVE X = POSITIVE 5/4.
SO THESE WOULD BE
THE TWO SOLUTIONS
TO OUR QUADRATIC EQUATION.
AND I SHOULD ALSO MENTION
WE COULD WRITE THE SOLUTION
AS X = + OR - 5/4.
THIS IS A SHORT WAY TO
REPRESENT BOTH POSITIVE 5/4
AND NEGATIVE 5/4.
LOOKING AT OUR SECOND EXAMPLE,
IF WE WANT TO SOLVE
THIS EQUATION BY FACTORING,
THE FIRST STEP IS TO SET
THE EQUATION EQUAL TO 0.
SO WE'LL START
BY SUBTRACTING 32
ON BOTH SIDES OF THE EQUATION.
ON THE LEFT SIDE, WE'D HAVE 2X
SQUARED - 32 = 32 - 32 IS 0.
AND THEN AGAIN IN THIS FORM,
IT DOESN'T LOOK LIKE WE HAVE
A DIFFERENCE OF SQUARES,
BUT AGAIN THE FIRST STEP
IS TO FACTOR OUT
THE GREATEST COMMON FACTOR
WHICH IN THIS WOULD BE 2.
IF WE FACTOR OUT 2,
WE'D HAVE 2 x THE QUANTITY
X SQUARED - 16 = 0.
X SQUARED IS A PERFECT SQUARE,
16 IS A PERFECT SQUARE,
AND WE HAVE A DIFFERENCE,
SO THIS DOES FACTOR
AS A DIFFERENCE OF SQUARES.
X x X IS EQUAL TO X SQUARED,
SO WE HAVE AN X HERE
AND AN X HERE.
AND 4 x 4 IS EQUAL TO 16,
SO WE HAVE +4 AND -4.
AND NOW BECAUSE THIS PRODUCT
IS EQUAL TO 0,
EITHER X + 4 MUST EQUAL 0
OR X - 4 MUST EQUAL 0.
SOLVING THIS EQUATION FOR X,
WE WOULD SUBTRACT 4
ON BOTH SIDES,
SO WE HAVE X = NEGATIVE 4.
OR HERE WE WOULD ADD 4
TO BOTH SIDES,
WE'D HAVE X = POSITIVE 4.
SO HERE ARE TWO SOLUTIONS
WHICH AGAIN IF WE WANTED TO,
WE COULD EXPRESS
AS X = + OR - 4.
SO THE MOST IMPORTANT THING
TO REMEMBER HERE
IS THE STEP IN FACTORING
IS ALWAYS TO FACTOR OUT
THE GREATEST COMMON FACTOR.
