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- 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.