- NOW WE'LL LOOK AT SOME EXAMPLES OF DETERMINING
THE CONSTANT TERM THAT MUST BE ADDED TO AN EXPRESSION
TO MAKE THE EXPRESSION A PERFECT SQUARE TRINOMIAL,
OR A TRINOMIAL THAT FACTORS INTO TWO EQUAL FACTORS.
AND THIS IS A SKILL THAT WE'LL NEED
IN ORDER TO SOLVE QUADRATIC EQUATIONS BY A METHOD
CALLED COMPLETING THE SQUARE.
LET'S START BY TAKING A LOOK AT THE NOTES DOWN HERE
AT THE BOTTOM.
IF WE HAVE A TRINOMIAL WITH A LEADING COEFFICIENT OF 1,
AND THE CONSTANT TERM IS B DIVIDED BY 2 SQUARED,
WHERE B IS THE COEFFICIENT OF THE MIDDLE TERM,
THIS WILL THEN BE A PERFECT SQUARE TRINOMIAL,
WHICH MEANS IT WILL FACTOR INTO TWO EQUAL FACTORS.
SO LOOKING AT OUR FIRST EXAMPLE WE NEED TO RECOGNIZE THAT B = 8,
SO B DIVIDED BY 2 SQUARED, WHICH IS B8 DIVIDED BY 2,
WHICH IS 4, SQUARED, WHICH IS EQUAL TO 16.
SO IF THE CONSTANT TERM HERE IS 16,
THIS WILL BE A PERFECT SQUARE TRINOMIAL,
AND LET'S VERIFY THIS BY FACTORING.
SO THIS WILL FACTOR INTO TWO BINOMIAL FACTORS.
THE FIRST TERMS MUST COME FROM THE FACTORS OF X SQUARED,
WHICH WOULD BE X AND X.
AND NOW WE WANT THE FACTORS OF 16 THAT ADD TO POSITIVE 8,
WHICH HAPPEN TO BE 4 AND 4.
SO WE HAVE 4 HERE AND 4 HERE.
NOTICE HOW WE HAVE TWO EQUAL BINOMIAL FACTORS,
WHICH MAKES THIS A PERFECT SQUARE TRINOMIAL FACTOR,
AND WE CAN WRITE THIS AS THE QUANTITY X + 4 SQUARED.
IN OUR NEXT EXAMPLE NOTICE HOW B = -14.
SO B DIVIDED BY 2 SQUARED IS NOW GOING TO BE -14 DIVIDED BY 2,
WHICH IS -7 SQUARED, WHICH EQUALS 49.
WHAT YOU'LL NOTICE IS THE CONSTANT TERM
IS ALWAYS GOING TO BE POSITIVE
IF WE HAVE A PERFECT SQUARE TRINOMIAL.
IF IT'S NEGATIVE, IT CANNOT BE A PERFECT SQUARE TRINOMIAL.
SO HERE WE HAVE 49, AND WE CAN GO AHEAD
AND FACTOR THIS TO VERIFY.
THE FACTORS OF X SQUARED, X AND X,
AND NOW WE WANT THE FACTORS OF 49 THAT HAVE A SUM OF -14,
SO NOW WE'LL HAVE TWO NEGATIVE FACTORS.
IT'S GOING TO BE -7 AND -7, SO WE HAVE -7 HERE AND -7 HERE.
AGAIN, WE HAVE TWO EQUAL FACTORS,
WHICH VERIFIES THIS IS A PERFECT SQUARE TRINOMIAL.
WE HAVE THE QUANTITY X - 7 SQUARED.
NOW, IN OUR LAST EXAMPLE NOTICE HOW B = -7.
SO WHEN B IS NEGATIVE, IT IS A LITTLE MORE INVOLVED.
SO AGAIN, B = -7, SO B DIVIDED BY 2 SQUARED,
NOTICE HOW -7 DIVIDED BY 2 = -7 HALVES,
SO THIS DOESN'T SIMPLIFY PERFECTLY AS IT DOES
WHEN B IS EVEN.
AND NOW WE'LL JUST SQUARE THE NUMERATOR
AND SQUARE THE DENOMINATOR.
-7 SQUARED IS 49, 2 SQUARED IS 4,
SO THIS IS THE CONSTANT TERM
THAT WE MUST ADD TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
AND NOW AGAIN WE WANT TO FACTOR THIS.
THIS IS WHY IT'S NICE TO SHOW THIS EXTRA WORK HERE.
THE FACTORS OF X SQUARED WOULD BE X AND X,
BUT DETERMINING THE FACTORS OF 49/4 THAT HAVE A SUM OF -7
IS NOT AN EASY QUESTION,
AND THAT'S THE REASON
WHY WE HAVE THIS SECOND LINE OF NOTES HERE.
THE CONSTANT TERM IN THE BINOMIAL FACTOR
IS B DIVIDED BY 2, OR THE NUMBER BEFORE IT'S SQUARED,
TO GET THIS CONSTANT TERM.
SO LOOKING BACK AT OUR WORK HERE, NOTICE HOW B/2 WAS -7/2,
WHICH MEANS WE'LL HAVE X - 7/2 HERE AS WELL AS HERE.
-7/2 x -7/2 = 49/4.
WE KNOW THAT FROM OUR WORK, BUT THE SUM IS ALSO -7.
SO TO FINISH, WE HAVE THE QUANTITY X - 7/2 SQUARED.
OKAY, I HOPE YOU FOUND THIS HELPFUL.