- WE WANT TO FIND 
THE X AND Y INTERCEPTS
OF THE QUADRATIC FUNCTION F 
OF X = X SQUARED +2X - 9.
TO FIND THE X INTERCEPTS 
OF ANY FUNCTION,
WE SET Y OR F OF X 
EQUAL TO ZERO AND SOLVE FOR X.
SO IF WE SUBSTITUTE ZERO 
FOR F OF X,
WE WOULD HAVE THE EQUATION 
0 = X SQUARED + 2X - 9,
BUT THIS DOES NOT FACTOR SO WE 
CAN'T SOLVE THIS BY FACTORING.
SO WE'LL HAVE TO SOLVE THIS
USING THE QUADRATIC FORMULA 
GIVEN HERE
WHERE "A" WOULD BE EQUAL TO 1, 
B WOULD BE EQUAL TO POSITIVE 2
AND C WOULD BE EQUAL TO -9.
SO OUR SOLUTIONS WOULD BE 
X = -B OR -2
PLUS OR MINUS 
THE SQUARE ROOT OF B SQUARED,
WHICH WOULD BE 2 SQUARED
- 4 x "A", WHICH IS 1 x C, 
WHICH IS -9
DIVIDED BY 2 x "A" OR 2 x 1.
SO THE TWO SOLUTIONS HERE 
WILL BE OUR X INTERCEPTS.
SO WE'D HAVE X EQUALS--
THIS WOULD BE -2 PLUS OR MINUS
THE SQUARE ROOT OF--THIS WOULD 
BE 4 AND THEN PLUS 36,
THAT'S THE SQUARE ROOT OF 40 
DIVIDED BY 2 X 1 or 2.
NEXT WE WANT TO SIMPLIFY 
THE SQUARE ROOT OF 40.
WE'RE LOOKING FOR THE PERFECT 
SQUARE FACTORS OF 40,
AND SINCE THE SQUARE ROOT 
OF 40
IS EQUAL TO THE SQUARE ROOT 
OF 4 x 10.
WELL, THE SQUARE ROOT OF 4 
IS EQUAL TO 2,
THIS SIMPLIFIES 
TO 2 SQUARE ROOT 10.
SO WE HAVE X = -2 +/- 2 
SQUARE ROOT 10 DIVIDED BY 2
WE NEED TO BE CAREFUL 
WHEN SIMPLIFYING THIS.
WE CANNOT JUST SIMPLIFY 
THESE 2s
BECAUSE WE CANNOT SIMPLIFY 
ACROSS ADDITION
OR SUBTRACTION.
SO TO HELP US SIMPLIFY THIS,
LET'S WRITE THIS AS 
TWO SEPARATE FRACTIONS.
WE HAVE -2 DIVIDED BY 2 +/- 2 
SQUARE ROOT 10 DIVIDED BY 2
AND NOW WE CAN SEE 
IT'S GOING TO SIMPLIFY TO -1
AND THEN JUST PLUS OR MINUS 
SQUARE ROOT 10.
AGAIN, THIS SIMPLIFIES TO ONE 
OVER ONE AND SO DOES THIS.
SO WE HAVE TWO SOLUTIONS HERE,
AND THEREFORE WE HAVE 
TWO X INTERCEPTS.
THE SMALLER X INTERCEPT WOULD 
BE -1 - SQUARE ROOT 10, 0
AND THE LARGER X INTERCEPT
WOULD BE -1 + SQUARE ROOT 
10, 0.
WE ARE GOING TO CHECK 
THESE POINTS GRAPHICALLY,
SO LET'S GO AHEAD 
AND GET DECIMAL APPROXIMATIONS
FOR BOTH OF THESE.
-1 - SQUARE ROOT 10 
IS APPROXIMATELY -4.16,
AND -1+ SQUARE ROOT 10 IS 
APPROXIMATELY POSITIVE 2.16.
SO AGAIN, THIS X INTERCEPT 
IS APPROXIMATELY -4.16, 0
AND THIS X INTERCEPT
IS APPROXIMATELY 
POSITIVE 2.16, 0.
WE DO WANT TO GIVE THE EXACT 
VALUES IN THE HOMEWORK,
BUT TO CHECK IT GRAPHICALLY
WE WOULD NEED THE DECIMAL 
APPROXIMATIONS.
NOW TO FIND THE Y INTERCEPT 
OF ANY FUNCTION,
WE SET X EQUAL TO ZERO 
AND SOLVE FOR Y OR F OF X.
SO IF X IS EQUAL TO ZERO, THE 
Y INTERCEPT WOULD BE F OF 0,
WHICH WOULD BE EQUAL TO 0 
SQUARED + 2 x 0 - 9,
WHICH WE CAN SEE 
WOULD BE EQUAL TO -9.
WHEN WE HAVE A QUADRATIC 
FUNCTION IN THIS FORM HERE,
THE VALUE OF C 
OR THE CONSTANT ON THE END
WILL ALWAYS BE 
THE Y INTERCEPT.
SO THE Y INTERCEPT 
WOULD BE THE POINT 0, -9.
LET'S LOOK AT THE GRAPH 
OF OUR FUNCTION
TO VERIFY THESE INTERCEPTS.
HERE IS THE SMALLER 
X INTERCEPT,
WHICH WE FOUND 
TO BE -1 - SQUARE ROOT 10, 0,
WHICH WAS APPROXIMATELY 
-4.16, 0.
HERE IS THE LARGER X INTERCEPT 
OF -1 + SQUARE ROOT 10, 0
OR APPROXIMATELY 2.16, 0,
AND FINALLY HERE 
IS THE Y INTERCEPT OF 0, -9.
I HOPE YOU FOUND THIS HELPFUL.
