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X
- WE WANT TO SOLVE THE GIVEN QUADRATIC EQUATION
BY COMPLETING THE SQUARE.
TO DO THIS, WE'LL BE FOLLOWING THE STEPS OUTLINED HERE.
I DO HAVE A MINI LESSON VIDEO ON COMPLETING THE SQUARE
THAT GOES THROUGH THESE IN MUCH MORE DETAIL,
OR YOU MAY WANT TO PAUSE THE VIDEO TO REVIEW
THESE BEFORE WE WORK OUT THE PROBLEM.
THE FIRST STEP HERE IS TO MOVE THE CONSTANT TERM
TO THE RIGHT SIDE OF THE EQUATION.
SO WE'LL HAVE TO SUBTRACT 17 ON BOTH SIDES OF THE EQUATION.
WE WOULD HAVE X SQUARED + 6X.
NOW WE'RE GOING TO MAKE THIS A PERFECT SQUARE TRINOMIAL
BY ADDING A CONSTANT HERE,
SO I'M GOING TO LEAVE A SPACE FOR THE CONSTANT.
THEN ON THE RIGHT SIDE OF THE EQUATION WE HAVE -17,
AND IF WE ADD A CONSTANT ON THE LEFT SIDE OF THE EQUATION,
WE'RE GOING TO HAVE TO ADD THE SAME CONSTANT TO THE RIGHT SIDE.
SO WE'LL ADD THE CONSTANT HERE AS WELL.
NOW THE NEXT STEP IS TO DETERMINE THE CONSTANT
THAT WE'RE GOING TO ADD HERE
SO THIS WILL BE A PERFECT SQUARE TRINOMIAL.
SO LET'S REVIEW THESE NOTES DOWN HERE AT THE BOTTOM.
IF WE HAVE A PERFECT SQUARE TRINOMIAL
WITH A LEADING COEFFICIENT OF 1,
THEN THE CONSTANT TERM IS GOING TO BE B DIVIDED BY 2 SQUARED,
WHERE B IS THE COEFFICIENT OF THE X TERM.
SO LOOKING AT OUR EXAMPLE, NOTICE THAT B = 6.
SO B DIVIDED BY 2 SQUARED IS GOING TO BE 6 DIVIDED BY 2,
THAT'S 3 SQUARED, WHICH IS EQUAL TO 9.
SO WE'LL HAVE TO ADD 9 HERE
TO MAKE THIS A PERFECT SQUARE TRINOMIAL.
AND OF COURSE IF WE ADD 9 TO THE LEFT SIDE OF THE EQUATION,
TO MAINTAIN EQUALITY, WE HAVE TO ADD 9 TO THE RIGHT SIDE AS WELL.
SO NOW WE'RE GOING TO FACTOR THIS TRINOMIAL
AND COMBINE THE TERMS ON THE RIGHT.
SO LET'S FACTOR THIS INTO TWO BINOMIAL FACTORS FIRST.
NOTICE THE RIGHT SIDE IS GOING TO BE -17 + 9, THAT'S -8.
AND NOW THE FIRST TERMS OF THE BINOMIAL FACTORS
WILL BE THE FACTORS OF X SQUARED, WHICH ARE X AND X.
NOW WE WANT THE FACTORS OF 9,
THAT HAVE A SUM OF 6, WHICH WOULD BE 3 AND 3.
SO WE HAVE PLUS 3 HERE AND PLUS 3 HERE.
SO NOTICE HOW WE HAVE TWO EQUAL FACTORS
VERIFYING THAT THIS IS A PERFECT SQUARE TRINOMIAL,
AND WE CAN WRITE THIS AS THE QUANTITY X + 3 SQUARED = -8.
AND NOW TO UNDO THIS SQUARING AND SOLVE FOR X,
WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION.
AND REMEMBER WHEN DOING SO,
WE'RE GOING TO HAVE A PLUS OR MINUS SIGN HERE,
AND NOW WHEN WE SIMPLIFY THE LEFT SIDE,
THIS WILL SIMPLIFY TO ONE FACTOR OF X + 3
AND ON THE RIGHT SIDE WE NEED TO SIMPLIFY THE SQUARE ROOT OF -8.
LET'S TAKE A LOOK AT THAT OVER HERE.
THE SQUARE ROOT OF -8 = THE SQUARE ROOT OF -1 x 2 x 2 x 2,
AND THIS SIMPLIFIES BECAUSE WE KNOW
THE SQUARE ROOT OF -1 = "I" AND THE SQUARE ROOT OF 2 SQUARED
OR 4 = 2.
SO THIS IS GOING TO SIMPLIFY TO 2I SQUARE ROOT OF 2.
LAST STEP HERE TO SOLVE FOR X,
WE'LL SUBTRACT 3 ON BOTH SIDES OF THE EQUATION.
SO WE HAVE X EQUALS--
THIS IS GOING TO BE -3 +/- 2I SQUARE ROOT OF 3.
SO REMEMBER WE HAVE TWO SOLUTIONS HERE,
ONE IS X = -3 + 2I SQUARE ROOT OF 3
AND THE OTHER SOLUTION IS X = -3 - 2I SQUARE ROOT OF 2.
SO IF WE WANT TO DESCRIBE OUR SOLUTIONS,
WE CAN SAY THAT WE HAVE TWO COMPLEX SOLUTIONS.
OKAY, I HOPE YOU FOUND THIS HELPFUL.