- WE WANT TO FACTOR AND SOLVE QUADRATIC EQUATIONS
IN WHICH A TRINOMIAL IS A PERFECT SQUARED TRINOMIAL,
WHERE IT FITS ONE OF THESE TWO FORMS.
ANOTHER -- OF MEMORIZING THESE TWO FORMULAS
BECAUSE IF WE LOOK AT OUR FIRST EXAMPLE,
WE HAVE X SQUARED + 16X + 64 = 0.
I RECOMMEND WE GO AHEAD AND FACTOR THIS AS WE NORMALLY WOULD
AND WE HAVE A LEADING COEFFICIENT OF 1.
SO IF THIS DOES FACTOR,
IT WOULD FACTOR INTO TWO BINOMIAL FACTORS
WHERE THE FIRST TERMS IN THE BINOMIAL FACTORS
WOULD COME FROM THE FACTORS OF X SQUARED,
WHICH SHOULD BE X AND X
AND THE SECOND TERMS IN THE BINOMIAL FACTORS
WOULD COME FROM THE FACTORS OF 64 THAT ADD TO 16.
WELL THE FACTORS OF 64 THAT ADD TO 16
JUST HAPPENS TO BE 8 AND 8
BECAUSE 8 x 8 = 64 AND 8 + 8 = 16.
SO THAT MEANS BOTH OF OUR BINOMIAL FACTORS WILL BE X + 8.
NOW BECAUSE OUR BINOMIAL FACTORS ARE THE SAME,
WE COULD WRITE THIS AS THE QUANTITY X + 8 SQUARED = 0.
THE ONLY WAY THE QUANTITY X + 8 SQUARED = 0
IS WHEN THE FACTOR OF X + 8 = 0.
SO IF WE SUBTRACT 8 ON BOTH SIDES OF THE EQUATION,
WE WILL HAVE OUR SOLUTION, WHICH IS X = -8.
NOW BECAUSE WE DO HAVE TWO FACTORS OF X + 8,
SOMETIMES WE WILL SAY THAT THIS SOLUTION
HAS MULTIPLICITY OF 2
OR THAT IT IS A DOUBLE ROOT OR DOUBLE ZERO.
LOOKING AT OUR SECOND EXAMPLE,
NOTICE HOW THE LEADING COEFFICIENT IS NOT EQUAL TO 1,
IT'S EQUAL TO 4 AND 4 IS NOT A COMMON FACTOR
AMONG THESE THREE TERMS.
SO NORMALLY, WHEN A IS NOT EQUAL TO 1
FACTORING A TRINOMIAL CAN BE A LITTLE BIT MORE CHALLENGING.
WE NOTICE THAT 4X SQUARED IS A PERFECT SQUARE
BECAUSE IT IS EQUAL TO 2X x 2X
AND THE THIRD TERM IS ALSO A PERFECT SQUARE
BECAUSE 9 x 9 = 81
AND THEREFORE IT LOOKS LIKE IT WILL SATISFY
THE FORMULAS ABOVE FOR A PERFECT SQUARE TRINOMIAL.
AGAIN, I DON'T LIKE TO MEMORIZE THIS.
SO WHAT I WOULD RECOMMEND IS
SEEING THAT THIS WILL FACTOR INTO TWO BINOMIAL FACTORS
AND FOR THE FACTORS OF 2X SQUARED, WE WOULD TRY 2X AND 2X,
AND THEN FOR THE FACTORS OF 81,
WE WOULD TRY 9 AND 9 AND THEN WE WILL USE
A SIGN OF THE MIDDLE TERM,
SO WE WILL USE A MINUS SIGN HERE AND MINUS SIGN HERE.
THEN TO SEE IF THIS ACTUALLY WORKS,
WE NEED YOU TO MAKE SURE THAT
THE SUM OF THE INNER PRODUCT AND THE OUTER PRODUCT
IS EQUAL TO THE MIDDLE TERM OF -36X.
NOTICE THE INNER PRODUCT IS -18X AND SO IS THE OUTER PRODUCT,
WHICH WOULD SUM TO -36X.
SO THIS IS FACTORED CORRECTLY.
SO NOTICE HOW WE HAVE A TRINOMIAL
TO RECOGNIZE THE FIRST TERM AND THE THIRD TERM
ARE PERFECT SQUARES,
WE CAN TRY TO FACTOR IN THIS WAY WITHOUT HAVING TO MEMORIZE
ANOTHER FACTORING FORMULA
AND AGAIN NOTICE HOW WE HAVE TWO EQUAL FACTORS HERE,
SO WE CAN WRITE THIS AS THE QUANTITY 2X - 9 SQUARED = 0.
AND THEREFORE THIS WILL ONLY EQUAL TO ZERO
WHEN THE FACTOR OF 2X - 9 = 0.
SO ADD 9 TO BOTH SIDES THAT WOULD GIVE US 2X = 9
AND THEN DIVIDE BOTH SIDES BY 2.
SO OUR SOLUTION IS X = 9 HALVES,
WHICH AGAIN BECAUSE WE HAVE TWO FACTORS OF 2X - 9,
WE CAN SAY THE SOLUTION HAS MULTIPLICITY OF 2
OR THAT IT IS A DOUBLE ROOT OR DOUBLE ZERO.
OKAY, HOPE YOU FOUND THIS HELPFUL.