- WE WANT TO FACTOR AND SOLVE THE GIVEN QUADRATIC EQUATIONS.
NOTICE IN BOTH CASES WE HAVE A TRINOMIAL
IN THE FORM OF AX SQUARED + BX + C = 0.
WHEN "A" IS NOT EQUAL TO 1, AND IT'S NOT A COMMON FACTOR.
WE'VE ACTUALLY COVERED THREE METHODS
FOR FACTORING THESE TYPES OF TRINOMIALS.
WE TALKED ABOUT THE TRIAL AND ERROR METHOD,
BOTTOMS UP METHOD, AS WELL AS THE GROUPING METHOD.
IN THIS VIDEO WE'LL REVIEW HOW TO FACTOR THESE
USING THE TRIAL AND ERROR METHOD.
SO FOR OUR FIRST EXAMPLE,
SINCE THERE ARE NO COMMON FACTORS AMONG THESE TERMS,
IF THIS DOES FACTOR,
IT WILL FACTOR INTO TWO BINOMIAL FACTORS.
AND WE KNOW THE FIRST TERMS OF THE BINOMIAL FACTORS
MUST COME FROM THE FACTORS OF 5X SQUARED.
SINCE 5 IS PRIME, IT HAS TO BE A 5X AND AN X.
NEXT WE KNOW THE SECOND POSITIONS MUST COME
FROM THE FACTORS OF +7
SO THAT THE SUM OF THE INNER PRODUCT AND OUTER PRODUCT
IS EQUAL TO +36X.
AGAIN, LUCKILY 7 IS PRIME
SO WE ONLY HAVE TO CONSIDER TWO FACTORS OF 7,
ONE IN ITSELF.
AGAIN NOTICE HOW WE WANT THE SUM OF THESE TWO PRODUCTS
TO EQUAL 36.
SO IF WE PUT A +7 HERE,
NOTICE HOW THAT WOULD GIVE US 35X.
THEN IF WE PUT THE +1 HERE,
NOTICE HOW THE INNER PRODUCT IS +1X
AND 35X + 1X DOES EQUAL 36X
WHICH MEANS THIS IS NOW FACTORED CORRECTLY.
AND BECAUSE THIS PRODUCT IS EQUAL TO ZERO,
THE FIRST FACTOR OF 5X + 1 MUST EQUAL 0
OR THE SECOND FACTOR OF X + 7 MUST EQUAL 0.
NOW TO FIND OUR SOLUTIONS,
WE'LL SOLVE THESE TWO EQUATIONS FOR X.
SO HERE WE'LL SUBTRACT ONE ON BOTH SIDES
SO WE HAVE 5X = -1, DIVIDE BOTH SIDES BY 5.
SO WE HAVE X = -1/5
OR HERE WE NEED TO SUBTRACT 7 ON BOTH SIDES
SO WE HAVE X = -7.
THESE WOULD BE OUR TWO SOLUTIONS
TO THE GIVEN QUADRATIC EQUATION.
NOW LET'S TRY ANOTHER ONE.
HERE WE HAVE 6X SQUARED - X - 12.
AND AGAIN THERE ARE NO COMMON FACTORS AMONG THESE TERMS
SO IF THIS DOES FACTOR,
IT WILL FACTOR INTO TWO BINOMIAL FACTORS
WHERE AGAIN THE FIRST TERMS MUST COME FROM THE FACTORS
OF 6X SQUARED.
SO IT COULD BE 6X AND X OR 3X AND 2X.
AGAIN USING THE TRIAL AND ERROR METHOD,
WE'RE NOT REALLY SURE.
LET'S GO AHEAD AND TRY 3X AND 2X.
THEN THE SECOND TERMS MUST COME FROM THE FACTORS OF -12.
SO THERE ARE QUITE A FEW FACTORS TO TRY
WE COULD USE -1 x 12, -12 x 1, -2 x 6,
-6 x 2, -3 x 4 OR -4 x 3.
BUT THE KEY IS IF WE WANT THE SUM
OF THE INNER PRODUCT AND OUTER PRODUCT TO BE EQUAL TO -1X.
SO AGAIN USING THE TRIAL AND ERROR METHOD,
WE'RE JUST GOING TO PLACE THESE FACTORS
INTO OUR BINOMIAL FACTORS
AND CHECK TO SEE WHICH ONES WILL WORK.
THERE ARE A COUPLE OF THINGS THAT WE CAN KEEP IN MIND
TO HELP US.
THESE BINOMIAL FACTORS CANNOT HAVE ANY COMMON FACTORS
SINCE THE ORIGINAL TRINOMIAL
DIDN'T HAVE ANY COMMON FACTORS.
SO WE WOULDN'T PUT A MULTIPLE OF TWO HERE
OR A MULTIPLE OF THREE HERE.
SO IF WE TRY PUTTING +3 HERE,
WE WOULD HAVE TO USE A -4 HERE.
AGAIN NOTICE HOW THESE BINOMIAL FACTORS
DON'T HAVE ANY COMMON FACTORS
AND NOW LET'S CHECK THE SUM OF THE INNER AND OUTER PRODUCT.
THE INNER PRODUCT IS -8X, THE OUTER PRODUCT IS +9X.
WELL THIS SUM WOULD BE +1X AND WE WANT -1X.
BUT THAT'S AN EASY FIX.
WE CAN USE A +4 HERE AND A -3 HERE
WHICH WOULD CHANGE THE SIGN OF THESE TWO TERMS
GIVING US THE SUM OF -1.
SO WE'LL CHANGE THIS TO PLUS
AND WE'LL CHANGE THIS TO MINUS.
AND AGAIN NOW THE INNER PRODUCT IS 8X,
THE OUTER PRODUCT IS -9X, WHICH WOULD HAVE A SUM OF -1X,
WHICH IS OUR MIDDLE TERM.
SO WE NOW KNOW THIS IS FACTORED CORRECTLY.
SO NOW WE CAN SOLVE THE EQUATION.
SINCE THIS PRODUCT IS EQUAL TO ZERO,
THE FACTOR OF 3X + 4 MUST EQUAL 0
OR THE FACTOR OF 2X - 3 MUST EQUAL 0.
AND NOW WE'LL SOLVE FOR X, FIRST UP HERE,
SO SUBTRACT 4 ON BOTH SIDES,
SO WE HAVE 3X = -4 DIVIDE BY 3,
WE HAVE X = -4/3
OR HERE WE WOULD ADD 3 TO BOTH SIDES.
IT SHOULD BE 2X = +3 AND DIVIDE BY 2.
OUR SECOND SOLUTION IS X = 3/2,
OKAY, HOPE YOU FOUND THIS HELPFUL.