- WE WANT TO WRITE EACH QUADRATIC EXPRESSION
IN THE FORM, THE QUANTITY X + "A" SQUARED + B,
WHICH MEANS WE'LL HAVE TO HAVE A PERFECT SQUARE TRINOMIAL
FOR THIS PART OF THE EXPRESSION
AND THEN WE'LL HAVE A CONSTANT ON THE END.
SO LOOKING AT OUR FIRST EXAMPLE,
WE'RE GOING TO ISOLATE THE X SQUARED AND X TERM
BY WRITING X SQUARED + 6X.
WE'RE GOING TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
SO WE'LL DETERMINE WHAT CONSTANT TO ADD HERE,
AND THEN WE STILL HAVE THE MINUS 4
AND THEN ON THE END WE'LL HAVE TO UNDO
WHATEVER CONSTANT WE ADD HERE
SO THAT THE EXPRESSIONS ARE STILL EQUAL.
SO WE'LL HAVE PLUS A CONSTANT HERE
AND THEN MINUS A CONSTANT HERE.
AND FOR A QUICK REVIEW, IF WE HAVE A QUADRATIC EXPRESSION
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,
THEN WE HAVE A PERFECT SQUARE TRINOMIAL
THAT WILL FACTOR INTO THESE TWO BINOMIAL FACTORS.
SO WE WANT TO ADD B DIVIDED BY 2 SQUARED HERE
TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
WELL, 6 DIVIDED BY 2 IS 3 AND 3 SQUARED IS 9.
SO WE NEED TO ADD 9 HERE
TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
BUT THEN TO UNDO THE PLUS NINE,
WE'LL SUBTRACT NINE ON THE END.
NOTICE THAT A POSITIVE 9 MINUS 9 WOULD BE ZERO.
SO NOW WE'RE GOING TO FACT THIS.
IT'LL BE A PERFECT SQUARE TRINOMIAL
AND THEN WE'LL COMBINE OUR LIKE TERMS HERE.
SO JUST TO SHOW THIS,
WE'LL FACTOR THIS FIRST INTO TWO BINOMIAL FACTORS.
WE CAN SEE THIS IS GOING TO BE MINUS 13.
SO WE HAVE X AND X, THE FACTORS OF X SQUARED.
THE FACTORS OF 9 THAT ADD TO 6 ARE 3 AND 3.
SO WE HAVE PLUS 3 HERE AND PLUS THREE HERE.
SO NOW WE CAN WRITE THIS AS A PERFECT SQUARE.
WE HAVE THE QUANTITY X + 3 SQUARED - 13.
NOTICE THIS EXPRESSION FITS THIS FORM HERE.
WE HAVE A BINOMIAL SQUARED PLUS OR MINUS A CONSTANT.
LET'S TRY ANOTHER EXAMPLE.
AGAIN, WE'LL ISOLATE THE X SQUARED AND X TERM.
SO WE'LL HAVE X SQUARED - 12X,
THEN WE'RE GOING TO ADD A CONSTANT
TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
WE'LL HAVE PLUS 80
AND THEN WE'LL SUBTRACT THE SAME CONSTANT.
AGAIN, THIS CONSTANT HERE HAS TO BE B DIVIDED BY 2 SQUARED.
HERE B IS -12.
-12 DIVIDED BY 2 IS -6, -6 SQUARED IS POSITIVE 36.
SO WE'LL HAVE PLUS 36 HERE AND MINUS 36 HERE.
NOW WE'LL FACTOR THIS AND COMBINE OUR CONSTANTS HERE.
80 - 36 IS EQUAL TO 44.
SO WE HAVE PLUS 44.
THE FACTORS OF X SQUARED, X AND X.
THE FACTORS OF 36 AND THE -12, -6 AND -6.
WE HAVE MINUS 6 HERE AND MINUS SIX HERE.
SO WE CAN WRITE THIS AS THE QUANTITY X - 6 SQUARED + 44.
NOW FOR OUR LAST EXAMPLE
IT'S GOING TO BE A LITTLE BIT MORE INVOLVED
BECAUSE NOTICE HOW B IS ODD.
B IS EQUAL TO -5.
SO WE'LL HAVE X SQUARED - 5X, PLUS A CONSTANT - 3,
MINUS A CONSTANT.
NOTICE HERE B IS EQUAL TO -5.
SO B DIVIDED BY 2 WOULD BE -5 DIVIDED BY 2 SQUARED.
SO HERE WE'RE GOING TO HAVE TO ADD--THIS WOULD BE 25/4.
SO WE'LL ADD 25/4 HERE AND SUBTRACT 25/4 HERE.
NOW THIS WILL BE A PERFECT SQUARE TRINOMIAL
AND WE'LL HAVE TO COMBINE THESE CONSTANTS.
NOW TO FACTOR THIS,
IT'S GOING TO BE HELPFUL TO REVIEW THE NOTES HERE.
THE CONSTANT TERM IN THE BINOMIAL FACTORS
IS B DIVIDED BY 2.
NOTICE HOW THAT'S THE NUMBER WE HAD HERE
BEFORE WE SQUARED TO GET OUR CONSTANT.
NOTICE IN THIS EXAMPLE, B DIVIDED BY 2 IS -5/2.
SO THAT MEANS WE'LL HAVE THE QUANTITY X - 5/2 SQUARED.
HERE -3 IS EQUAL TO -12/4,
SO WE'LL HAVE MINUS 12/4 - 25/4.
SO WE HAVE THE QUANTITY X - 5/2 SQUARED,
THIS IS GOING TO BE MINUS 37/4.
SO IT IS A LITTLE BIT MORE INVOLVED WHEN B IS ODD.