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- WE WANT TO SOLVE THE GIVEN QUADRATIC EQUATION
BY COMPLETING THE SQUARE.
THIS VIDEO DOES ASSUME THAT YOU'RE FAMILIAR
WITH THE BASIC STEPS FOR COMPLETING THE SQUARE,
OUTLINED HERE IN THESE FIVE STEPS,
WHICH I DID COVER IN THE MINI LESSON
ON COMPLETING THE SQUARE.
SO IF YOU DIDN'T WATCH THAT VIDEO,
YOU MAY WANT TO OR AT LEAST REVIEW THESE FIVE STEPS.
GOING BACK TO OUR EXAMPLE,
THIS IS GOING TO BE A LITTLE BIT MORE INVOLVED
BECAUSE NOTICE HOW OUR LEADING COEFFICIENT IS EQUAL TO THREE.
THE FIRST STEP HERE WILL BE TO MOVE THE CONSTANT TERM
TO THE RIGHT SIDE OF THE EQUATION.
SO TO UNDO MINUS 2,
WE'LL ADD 2 TO BOTH SIDES OF THE EQUATION.
SO WE'LL HAVE 3X SQUARED - 7X.
NOW WE ARE GOING TO MAKE THIS A PERFECT SQUARE TRINOMIAL,
SO WE'LL HAVE TO ADD A CONSTANT HERE AT SOME POINT.
LET'S PUT PLUS BLANK TO REPRESENT A CONSTANT EQUALS--
AND THEN ON THE RIGHT SIDE WE'LL HAVE TWO,
AND THEN TO MAINTAIN EQUALITY
WE'LL HAVE TO ADD THE SAME CONSTANT ON THE RIGHT.
SO WE DO WANT TO MAKE THIS A PERFECT SQUARE TRINOMIAL,
BUT BEFORE WE DO THIS, WE NEED THE LEADING COEFFICIENT HERE
TO BE EQUAL TO ONE.
SO WHAT WE'LL HAVE TO DO IS DIVIDE EVERY TERM BY 3.
SO WE'LL DIVIDE THIS BY 3, DIVIDE THIS BY 3
AND DIVIDE THIS BY 3.
SO NOW WE'LL HAVE X SQUARED - 7/3X PLUS A CONSTANT = 2/3
PLUS A CONSTANT.
NOW, LET'S LOOK AT OUR NOTES DOWN HERE AT THE BOTTOM.
IF WE HAVE A QUADRATIC TRINOMIAL, AS WE DO HERE,
WITH A LEADING COEFFICIENT OF ONE,
THE CONSTANT TERM NEEDS TO BE B DIVIDED BY 2 SQUARED,
WHERE B IS THE COEFFICIENT OF THE X TERM
IN ORDER FOR THIS TO BE A PERFECT SQUARE TRINOMIAL.
NOW REMEMBER, DIVIDING BY 2
IS THE SAME AS MULTIPLYING BY 1/2.
SO IT'S PROBABLY GOING TO BE MORE CONVENIENT
IF WE USE 1/2 x B SQUARED
SINCE IN THIS PROBLEM B IS THE FRACTION -7/3.
SO LET'S DETERMINE WHAT THE CONSTANT NEEDS TO BE.
WE'RE GOING TO HAVE 1/2 x -7/3
AND THEN WE'RE GOING TO SQUARE THIS.
SO THIS IS GOING TO BE -7/6 SQUARED,
WHICH IS GOING TO BE EQUAL TO 49/36.
WE NEED TO ADD 49/36 HERE
TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
AND IF WE ADD THIS TO THE LEFT SIDE OF THE EQUATION,
WE MUST DO THE SAME TO THE RIGHT SIDE.
NOW WE'RE GOING TO FACTOR THIS,
WHICH WILL BE A PERFECT SQUARE TRINOMIAL
AND THEN WE'LL ADD THESE TERMS ON THE RIGHT.
THIS CAN BE DIFFICULT TO FACTOR
AND THAT'S WHY I HAVE THESE EXTRA NOTES HERE
THAT SHOW THE CONSTANT TERM OF THE BINOMIAL FACTOR,
WHICH IS B DIVIDED BY 2, OR IN OUR CASE, 1/2B.
IT'S THE NUMBER WE HAD HERE BEFORE WE SQUARED.
SO LOOKING AT OUR WORK HERE,
THE CONSTANT TERM IN THE BINOMIAL FACTOR
IS GOING TO BE -7/6,
WHICH MEANS THIS WILL FACTOR INTO TWO BINOMIAL FACTORS.
THE FIRST TERMS WILL BE X AND X, THE FACTORS OF X SQUARED.
AND THEN THE FACTORS OF 49/36 THAT ADD TO -7/3
ARE GOING TO BE -7/6 AND -7/6.
SO I'LL HAVE MINUS 7/6 HERE AND MINUS 7/6 HERE.
NOW ON THE RIGHT, IN ORDER TO ADD THESE FRACTIONS
WE DO HAVE TO HAVE A COMMON DENOMINATOR,
WHICH WOULD BE 36.
SO I'LL MULTIPLY THIS FACTION BY 12/12.
SO ON THE RIGHT SIDE WE'RE GOING TO HAVE 24/36 + 49/36.
SO WE CAN WRITE THIS
AS THE QUANTITY X -7/6 SQUARED EQUALS--
THIS IS GOING TO BE 73/36.
AND NOW WE CAN UNDO THE SQUARING AND SOLVE FOR X
BY TAKING THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION.
SO I'LL TAKE THE SQUARE ROOT OF THE LEFT SIDE,
THE SQUARE ROOT OF THE RIGHT SIDE--
REMEMBER WE DO HAVE TO HAVE A PLUS OR MINUS SIGN HERE
TO OBTAIN ALL SOLUTIONS.
NOW ON THE LEFT SIDE WE HAVE ONE FACTOR
OF X - 7/6 = PLUS OR MINUS--
AND NOW TO SIMPLIFY THIS,
REMEMBER WHEN WE HAVE A FRACTION
UNDERNEATH THE SQUARE ROOT,
WE CAN SQUARE ROOT THE NUMERATOR AND DENOMINATOR
SEPARATELY.
NOTICE HOW THE DENOMINATOR SIMPLIFIES PERFECTLY TO 6
AND THE NUMERATOR DOES NOT SIMPLIFY.
AND NOW TO SOLVE FOR X WE'LL ADD 7/6
TO BOTH SIDES OF THE EQUATION.
SO WE WOULD HAVE X = 7/6 +/- SQUARE ROOT 73, ALL OVER 6.
AND BECAUSE THE SQUARE ROOT 73,
WE CAN TELL WE HAVE TWO REAL IRRATIONAL SOLUTIONS.
KEEP IN MIND WE DO HAVE TWO SOLUTIONS HERE,
ONE WHERE WE ADD SQUARE ROOT 73 OVER 6
AND ONE WHERE WE SUBTRACT THE SQUARE ROOT 73 OVER 6.
AND SOMETIMES YOU WILL SEE THIS EXPRESSED
AS A SINGLE FRACTION,
SINCE WE DO HAVE A COMMON DENOMINATOR.
WE COULD WRITE THIS AS X EQUALS A NUMERATOR
OF 7 +/- SQUARE ROOT 73
ALL OVER OUR COMMON DENOMINATOR OF 6.
EITHER FORM IS ACCEPTABLE.
JUST KEEP IN MIND THIS DOES REPRESENT TWO SOLUTIONS.
OKAY. I HOPE YOU FOUND THIS HELPFUL.