- NOW, WE'LL LOOK AT SEVERAL EXAMPLES
OF SOLVING QUADRATIC EQUATIONS GRAPHICALLY.
WE WANT TO USE THE GRAPH OF Y = X SQUARED + X - 6,
PROVIDED, HERE IN RED,
TO SOLVE THE EQUATION X SQUARED + X - 6 = 0.
IF WE COMPARE THESE TWO EQUATIONS,
NOTICE THAT Y HAS BEEN REPLACED WITH 0.
WHEN WE'RE GIVEN THE EQUATION OF A FUNCTION,
REPLACING Y WITH 0 AND SOLVING FOR X
IS THE PROCESS FOR FINDING THE X-INTERCEPTS.
SO WE CAN SOLVE THIS EQUATION HERE GRAPHICALLY
BY FINDING THE X-INTERCEPTS OF OUR FUNCTION
WHICH OCCUR HERE AND HERE WHERE THE FUNCTION CROSSES THE X-AXIS.
NOTICE THE COORDINATES OF THIS X-INTERCEPT WOULD BE (2,0)
AND THE COORDINATES OF THIS X-INTERCEPT WOULD BE (-3,0).
NOTICE FOR THIS X-INTERCEPT,
THIS IS TELLING US THAT WHEN X = 2, Y = 0,
SO X = 2 WOULD BE A SOLUTION TO THIS EQUATION.
OR WHEN X = -3, Y = 0, AND THEREFORE X = -3
WOULD BE A SOLUTION TO OUR EQUATION.
SO IN GENERAL WHEN WE HAVE AN EQUATION
EQUAL TO 0 AND WE REPLACE 0 WITH Y AND GRAPH THE FUNCTION,
THE SOLUTIONS WILL BE THE X-INTERCEPTS.
NOTICE FOR THIS QUADRATIC EQUATION,
WE'VE TWO REAL RATIONAL SOLUTIONS.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
HERE WE WANT TO USE THE GRAPH OF Y = -X SQUARED + 4
TO SOLVE THE EQUATION - X SQUARED + 4 = 0.
SO BECAUSE THIS EQUATION IS EQUAL TO 0,
THE SOLUTIONS WILL BE
THE X- INTERCEPTS OF Y = -X SQUARED + 4.
SO AGAIN, WE HAVE TWO SOLUTIONS, ONE HERE AND ONE HERE.
THE COORDINATES OF THIS X-INTERCEPT WOULD BE (2,0).
THE COORDINATES OF THIS X-INTERCEPT WOULD BE (-2,0).
AGAIN, THESE ARE THE X VALUES
WHERE Y = 0 WHICH WOULD BE A SOLUTION TO THIS EQUATION.
SO OUR TWO SOLUTIONS ARE X = 2 OR X = -2.
AND AGAIN, WE HAVE TWO REAL RATIONAL SOLUTIONS.
IN THIS EXAMPLE, WE WANT TO USE THE GRAPH OF Y = X SQUARED - 2X
+ 1 TO SOLVE X SQUARED - 2X + 1 = 0.
AND NOTICE THIS GRAPH IS A LITTLE BIT DIFFERENT.
IT DOESN'T PASS THROUGH THE X-AXIS.
THE GRAPH ONLY TOUCHES THE X- AXIS AT X = 1.
THIS IS WHAT HAPPENS WHEN THE SOLUTION HAS MULTIPLICITY 2
OR WE CAN SAY IS A DOUBLE 0.
SO THE X-INTERCEPT HERE IS THE (.1,0),
SO OUR SOLUTION IS X = 1.
BUT AGAIN BECAUSE IT TOUCHES THE X-AXIS
AND DOESN'T PASS THROUGH IT,
WE CAN SAY THE SOLUTION HAS MULTIPLICITY OF 2.
WE CAN ALSO SAY IT'S A DOUBLE ZERO OR A DOUBLE ROOT.
SO IN THIS CASE, WE ONLY HAVE ONE REAL RATIONAL SOLUTION.
ANOTHER WAY TO SEE WHY WE SAY THIS HAS MULTIPLICITY OF 2,
IF WE ACTUALLY TRY TO SOLVE THIS BY FACTORING,
IT WOULD FACTOR INTO TWO BINOMIAL FACTORS.
WE'D WANT THE FACTORS OF 1 THAT ADD THE -2,
SO WE'D HAVE X - 1 x X - 1.
NOTICE HOW THE FACTOR OF X - 1 OCCURS TWICE,
WHICH IS ALSO ANOTHER REASON WHY WE CAN SAY
X = 1 HAS MULTIPLICITY OF 2.
WE HAVE 2 FACTORS OF X - 1.
BUT THIS IS HOW IT LOOKS GRAPHICALLY.
IN OUR LAST EXAMPLE,
WE WANT TO USE THE GRAPH OF Y = X SQUARED + 1
TO SOLVE X SQUARED + 1 = 0.
BUT IN THIS EXAMPLE,
NOTICE HOW WE'RE LOOKING FOR THE X-INTERCEPTS.
BUT THIS GRAPH DOES NOT HAVE ANY X-INTERCEPTS,
SO THAT MEANS THERE ARE NO REAL SOLUTIONS TO THIS EQUATION.
BUT IF WE TRIED TO SOLVE THIS ALGEBRAICALLY,
WE COULD ISOLATE THE X SQUARED BY SUBTRACTING 1 ON BOTH SIDES.
THIS WOULD GIVE US X SQUARED = -1.
IF WE TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION,
REMEMBER WE NEED A PLUS OR MINUS SIGN HERE,
WE WOULD HAVE X = PLUS OR MINUS,
WELL, THE SQUARE ROOT OF -1 IS EQUAL TO "I".
SO NOW TO BE MORE SPECIFIC ABOUT THE TYPES OF SOLUTIONS
WE HAVE HERE, WE COULD SAY THAT THIS EQUATION
HAS TWO IMAGINARY SOLUTIONS
AND THAT'S THE REASON WHY THERE ARE NO X-INTERCEPTS.
THE X-AXIS IS A REAL NUMBER LINE,
THEREFORE IF OUR QUADRATIC EQUATION
DOES NOT HAVE ANY REAL SOLUTIONS,
THE CORRESPONDING QUADRATIC FUNCTION
WILL NOT HAVE ANY X-INTERCEPTS.
OKAY, I HOPE YOU FOUND THIS HELPFUL.