- WE WANT TO SOLVE THE FOLLOWING QUADRATIC EQUATIONS
BY FACTORING.
NOTICE IN THE FIRST TWO EXAMPLES WE HAVE A TRINOMIAL WHEN A = 1.
AND SINCE THERE ARE NO COMMON FACTORS HERE, IF THIS FACTORS,
IT WILL FACTOR INTO TWO BINOMIAL FACTORS,
WHERE THE FIRST TERMS WILL BE THE FACTORS OF X SQUARED,
WHICH ARE X AND X.
AND NOW WE WANT THE FACTORS OF C THAT ADD TO B,
OR THE FACTORS OF 4 THAT ALSO ADD TO 4.
WELL, 4 DOES HAPPEN TO BE A PERFECT SQUARE,
AND 2 x 2 = 4, AND 2 + 2 = 4.
SO THE TWO FACTORS OF 4 THAT WE WANT ARE 2 AND 2.
SO NOTICE THAT BOTH FACTORS WILL BE X + 2.
SO IT IS TRUE THAT THIS IS A PERFECT SQUARE TRINOMIAL,
BUT WE DID FACTOR IT AS WE NORMALLY DO WHEN "A" = 1.
BUT NOW, SINCE WE DO HAVE TWO EQUAL FACTORS,
WE CAN WRITE THIS AS THE QUANTITY X + 2 SQUARED = 0.
SO THIS EQUATION WILL ONLY BE 0 WHEN X + 2 = 0.
SO IF WE SUBTRACT 2 ON BOTH SIDES,
WE KNOW OUR SOLUTION IS X = -2.
NORMALLY WHEN SOLVING A QUADRATIC EQUATION,
WE DO EXPECT TWO SOLUTIONS.
BUT SINCE WE HAVE TWO FACTORS OF X + 2,
WE DO ONLY HAVE ONE SOLUTION.
BUT WE CAN SAY THAT IT HAS MULTIPLICITY OF 2,
OR THAT IT'S A DOUBLE ROOT, OR DOUBLE 0.
WE'LL APPROACH THE SECOND EXAMPLE IN THE SAME WAY.
THERE ARE NO COMMON FACTORS.
IF THIS FACTORS, IT WILL FACTOR INTO TWO BINOMIAL FACTORS.
THE FIRST TERMS COME FROM THE FACTORS OF X SQUARED,
WHICH ARE X AND X.
AND NOW WE WANT THE FACTORS OF 25 THAT ADD TO -10.
AGAIN, NOTICE THAT 25 IS A PERFECT SQUARE,
SINCE 5 x 5 = 25.
BUT SINCE WE WANT A SUM OF -10,
IT'S ALSO TRUE THAT -5 x -5 = 25, AND THAT -5 + -5 = -10.
SO NOW WE'RE GOING TO HAVE TWO FACTORS OF X - 5.
SO AGAIN, WE HAVE A PERFECT SQUARE TRINOMIAL,
SO WE CAN REWRITE THIS AS THE QUANTITY X - 5 SQUARED = 0.
THEREFORE, THIS = 0 ONLY WHEN X - 5 = 0.
SO IF WE ADD 5 TO BOTH SIDES, WE HAVE ONE SOLUTION OF X = 5.
AGAIN, BECAUSE THERE ARE TWO FACTORS OF X - 5,
WE CAN SAY THIS SOLUTION HAS A MULTIPLICITY OF 2,
OR THAT IT'S A DOUBLE 0.
NOW, LOOKING AT THE LAST EXAMPLE,
NOTICE HOW THERE ARE NO COMMON FACTORS,
BUT "A" DOES NOT = 1.
THIS IS WHEN IT'S HELPFUL TO RECOGNIZE THAT THE FIRST TERM
ON THIS TRINOMIAL IS A PERFECT SQUARE,
AND SO IS THE LAST TERM.
WE KNOW THIS, BECAUSE WE CAN WRITE 4X SQUARED
AS 2X TO THE SECOND,
AND WE CAN WRITE 9 AS 3 SQUARED.
SO MY SUGGESTION IS WHEN WE NOTICE THIS,
LET'S GO AHEAD AND TRY TO FACTOR THIS
AS A PERFECT SQUARE TRINOMIAL,
MEANING WE'LL TRY TO FACTOR THIS INTO TWO BINOMIALS.
THE FIRST TERMS COME FROM THE FACTORS OF 4X SQUARED,
AND WE'LL USE 2X AND 2X.
THE SECOND TERMS COME FROM THE FACTORS OF 9,
AND WE'LL USE 3 AND 3.
AND WE'LL USE THE SIGN OF THE MIDDLE TERM,
SO WE'LL HAVE MINUS HERE AND MINUS HERE.
NOW, TO MAKE SURE THIS ACTUALLY WORKS,
WE'LL CHECK THE SUM OF THE INNER PRODUCT AND THE OUTER PRODUCT,
TO MAKE SURE WE DO HAVE -12X.
NOTICE HOW WE HAVE -6X HERE AND -6X HERE,
WHICH WOULD HAVE A SUM OF -12X.
SO IT IS A PERFECT SQUARE TRINOMIAL,
AND WE FACTORED IT CORRECTLY.
SO WE CAN WRITE THIS AS THE QUANTITY 2X - 3 SQUARED = 0,
WHICH MEANS THIS WILL = 0 WHEN 2X - 3 = 0.
SO IF WE ADD 3 TO BOTH SIDES, DIVIDE BY 2,
OUR SOLUTION IS X = 3/2.
AND AGAIN, WE CAN SAY THIS HAS A MULTIPLICITY OF 2,
OR THAT IT'S A DOUBLE 0.
NOTICE HOW IN THESE EXAMPLES
I DIDN'T APPLY THIS FORMULA DIRECTLY.
I'M NOT A BIG FAN OF MEMORIZING FORMULAS.
BUT IT IS IMPORTANT TO RECOGNIZE
THAT IF WE DO HAVE A PERFECT SQUARE TRINOMIAL,
THE FIRST TERM AND THE THIRD TERM WOULD BE PERFECT SQUARES.
AND THE PART THAT I DON'T WORRY TOO MUCH ABOUT
IS THAT THE MIDDLE TERM WOULD BE 2 x A x B,
BECAUSE WE CHECKED THE SUM OF THE INNER AND OUTER PRODUCT
TO MAKE SURE WE DID COME UP WITH THE MIDDLE TERM.
OKAY. I HOPE YOU FOUND THIS HELPFUL.