- IN THIS EXAMPLE WE'RE ASKED 
TO SOLVE THE QUADRATIC EQUATION
49 SQUARED - 36 = 0
USING THE QUADRATIC FORMULA.
HOPEFULLY WE RECOGNIZE THAT 
WE COULD SOLVE THIS BY FACTORING
SINCE THIS IS A DIFFERENCE 
OF SQUARES.
BUT WE'LL GO AHEAD AND SOLVE 
THIS USING THE QUADRATIC FORMULA
AND THEN CHECK IT BY FACTORING.
SO THE FIRST THING WE SHOULD 
RECOGNIZE IS THAT
THE EQUATION 
IS IN THE CORRECT FORM
OR THE FORM 
AX SQUARED + BX + C = 0,
THEREFORE A = +49, 
B WOULD BE ZERO, AND C = -36.
SO, AGAIN, A = 49, B = 0 
SINCE THERE'S NO X TERM,
AND C = -36.
NOW WE'LL PERFORM SUBSTITUTION 
INTO THE QUADRATIC FORMULA.
SO WE'LL HAVE X = -B 
THAT'S GOING TO BE -0
PLUS OR MINUS THE SQUARE ROOT 
OF B SQUARED
THAT'S ZERO SQUARED - 4 x A 
WHICH IS 49 x C WHICH IS -36.
ALL DIVIDED BY 2 x A OR 2 x 49.
SO NOW WE'LL BEGIN TO SIMPLIFY.
WE'LL HAVE X EQUALS--
THIS IS JUST ZERO
SO WE HAVE PLUS OR MINUS 
THE SQUARE ROOT OF--
THIS WOULD BE -4 x 49 x 36
WHICH IS ACTUALLY EQUAL TO 7,056 
DIVIDED BY 2 x 49 WHICH IS 98.
NOW IT'S DIFFICULT TO RECOGNIZE, 
BUT 7,056 IS A PERFECT SQUARE.
IF WE TAKE THIS SQUARE ROOT 
OF 7,056,
COMES OUT PERFECTLY TO 84.
WHICH MEANS X IS EQUAL TO 
PLUS OR MINUS 84 DIVIDED BY 98.
AND THIS FRACTION DOES SIMPLIFY.
84 IS EQUAL TO 6 x 14 
AND 98 IS EQUAL TO 7 x 14.
WELL, OF COURSE 14/14 
SIMPLIFIES TO 1.
SO WE HAVE X = 
PLUS OR MINUS 6/7.
OR, IF WE WANT, WE CAN SAY 
X = +6/7 OR X = -6/7.
AND BECAUSE WE HAVE TWO REAL 
RATIONAL SOLUTIONS
THIS DOES TELL US THAT 
THE ORIGINAL EQUATION
WAS FACTORABLE.
SO TO CHECK OUR SOLUTIONS LET'S 
GO AND SOLVE THIS BY FACTORING.
49X SQUARED - 36 IS 
THE DIFFERENCE OF SQUARES.
SO IT'LL FACTOR INTO 
TWO BINOMIAL FACTORS.
49X SQUARED IS EQUAL TO 7X x 7X.
36 IS A PERFECT SQUARE 
SINCE IT'S EQUAL TO 6 x 6.
AND ONE BINOMIAL WILL BE A SUM.
ONE BINOMIAL WILL BE 
A DIFFERENCE.
SO IF THIS PRODUCT IS EQUAL TO 
ZERO THEN 7X + 6 MUST EQUAL ZERO
OR 7X - 6 MUST EQUAL ZERO.
HERE WE WOULD SUBTRACT 6 
ON BOTH SIDES, DIVIDE BY 7.
HERE'S ONE SOLUTION OF X = -6/7.
AND OVER HERE WE WOULD ADD 6 
TO BOTH SIDES, DIVIDE BY 7.
WE HAVE OUR SECOND SOLUTION 
OF X = 6/7.
SO, AGAIN, 
NORMALLY WHEN GIVEN A CHOICE,
IF WE CAN FACTOR THE QUADRATIC 
WE WOULD SOLVE IT BY FACTORING
BUT REMEMBER THE QUADRATIC 
FORMULA WILL SOLVE ANY QUADRATIC
WHETHER THE SOLUTIONS ARE REAL, 
RATIONAL, IRRATIONAL,
OR EVEN COMPLEX.
OKAY, I HOPE YOU FOUND THIS 
HELPFUL.
