- 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.
