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- WE WANT TO WRITE THE GIVEN EQUATION IN QUADRATIC FORM
AND THEN SOLVE BY FACTORING.
SO FOR THIS EQUATION, QUADRATIC FORM WOULD BE THE FORM
U SQUARED - 9U + 8 = 0.
SO THE MAIN IDEA HERE IS WE WANT TO SEE
IF WE SQUARE THE VARIABLE FACTOR OF THE MIDDLE TERM,
IN THIS CASE X TO THE 3rd,
IT WOULD EQUAL THE VARIABLE FACTOR OF THE 1st TERM
WHICH IS X TO THE 6th.
SO WE'RE GOING TO WRITE THIS AS X TO THE 3rd,
THE VARIABLE FACTOR OF THE MIDDLE TERM SQUARED,
AND SEE IF IT DOES EQUAL THE VARIABLE FACTORS
OF THE 1ST TERM.
AND THIS IS EQUAL X TO THE 6th,
SO WE CAN WRITE THIS IN QUADRATIC FORM.
WE WOULD X TO THE 3rd SQUARED - 9 x X TO THE 3rd + 8 = 0.
SO IN THIS CASE, WE WOULD HAVE U = X TO THE 3rd,
BUT I DON'T THINK IT'S NECESSARY TO PERFORM THAT SUBSTITUTION
IF WE RECOGNIZE THIS IS IN QUADRATIC FORM HERE.
WE'LL GO AHEAD AND FACTOR THIS,
BUT NOW INSTEAD OF WRITING X AND X,
WE'RE GOING TO WRITE X TO THE 3rd AND X TO THE 3rd.
AND NOW, WE'LL FIND THE FACTORS OF +8 THAT ADD TO -9,
THAT WOULD BE -8 AND -1.
AND NOW, THIS PRODUCT IS EQUAL TO 0
WHEN THESE FACTORS ARE EQUAL TO 0,
SO WE'LL HAVE X TO THE 3rd - 8 = 0
OR X TO THE 3rd - 1 = 0.
SO HERE WE'LL ADD 8 TO BOTH SIDES OF THE EQUATION,
AND HERE WE'LL ADD 1 TO BOTH SIDES OF THE EQUATION.
AND NOW, WE WANT X, NOT X TO THE 3rd,
SO WE'LL TAKE THE CUBE ROOT OF BOTH SIDES OF THE EQUATION.
SO HERE WE'LL HAVE X EQUALS--
THE CUBE ROOT OF 8 IS EQUAL TO 2 SINCE 8 IS EQUAL TO 2 x 2 x 2,
SO WE HAVE X = 2 AND THE CUBE ROOT OF 1 IS JUST 1.
SO IT APPEARS WE HAVE TWO SOLUTIONS,
BUT IT IS IMPORTANT THAT WE GO AHEAD AND CHECK THESE
TO MAKE SURE THEY BOTH WORK.
SO LET'S TRY X = 2, FIRST.
IF X IS 2,
WE WOULD HAVE 2 TO THE 6th - 9 x 2 TO THE 3rd + 8 = 0.
WELL, 2 TO THE 6th IS EQUAL TO 8 x 8 OR 64,
2 TO THE 3rd IS 8, SO 9 x 8 IS 72,
SO 64 - 72 + 8 IS EQUAL TO 0,
SO THAT SOLUTION CHECKS.
AND NOW, WE'LL CHECK X = 1.
WE WOULD HAVE 1 TO THE 6th - 9 x 1 TO THE 3rd + 8 = 0,
WE'D HAVE 1 - 9 + 8 WHICH DOES EQUAL 0,
SO THAT SOLUTION CHECKS AS WELL.
SO WE HAVE 2 SOLUTIONS TO THIS EQUATION, X = 2 OR X = +1.
OKAY, WE'LL TAKE A LOOK AT ONE MORE EXAMPLE
IN THE NEXT VIDEO.