- NOW, WE'LL LOOK AT SOME EXAMPLES OF FACTORING
AND SOLVING QUADRATIC EQUATIONS
IN THE FORM X SQUARED + BX + C = 0.
NOTICE HOW THE LEADING COEFFICIENT IS EQUAL TO 1.
OUR FIRST EQUATION, WE HAVE X SQUARED + 4X - 21 = 0.
SO IF THIS TRINOMIAL FACTORS,
IT WILL FACTOR INTO TWO BINOMIAL FACTORS
WHERE THE TERMS IN THE FIRST POSITIONS
WILL COME FROM THE FACTORS OF X SQUARED WHICH WOULD BE X AND X,
AND THE TERMS IN THE SECOND POSITIONS
WILL BE THE FACTORS OF C THAT ADD TO B,
OR IN THIS CASE THE FACTORS OF NEGATIVE 21
THAT ADD TO POSITIVE 4.
LET'S LIST ALL THE FACTORS THAT HAVE NEGATIVE 21.
WE COULD HAVE NEGATIVE 1 x 21
OR NEGATIVE 21 x 1 OR NEGATIVE 3 x 7 OR NEGATIVE 7 x 3.
WE WANT THE TWO FACTORS THAT HAVE A SUM OF POSITIVE 4,
WHICH MEANS THE WINNING FACTORS
WILL BE NEGATIVE 3 AND POSITIVE 7.
THEREFORE ONE OF THE BINOMIAL FACTORS WILL BE X - 3
AND THE OTHER FACTOR WILL BE X + 7.
BECAUSE THE 3 IS NEGATIVE, WE HAVE -3.
BECAUSE THE 7 IS POSITIVE, WE HAVE 7.
OF COURSE IF WE PUT THE 7 HERE AND THE -3 HERE,
IT'S NOT GOING TO CHANGE OUR SOLUTIONS.
SO NOW, WE HAVE THE TRINOMIAL IN FACTORED FORM
WHERE THIS PRODUCT IS EQUAL TO ZERO,
WE CAN USE THE ZERO PRODUCT PROPERTY TO SOLVE THIS EQUATION.
AGAIN IF THIS PRODUCT IS EQUAL TO 0,
EITHER THE FIRST FACTOR OF X - 3 MUST EQUAL 0
OR THE SECOND FACTOR OF X + 7 MUST EQUAL 0.
NOW, WE'LL SOLVE EACH OF THESE EQUATIONS FOR X.
SO HERE WE WOULD ADD 3 TO BOTH SIDES.
SO WE'D HAVE X = 3, OR HERE WE WOULD SUBTRACT 7 ON BOTH SIDES,
SO WE'D HAVE X = NEGATIVE 7.
THESE ARE THE TWO SOLUTIONS TO OUR QUADRATIC EQUATION.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
AGAIN IF THIS DOES FACTOR, IT'LL FACTOR INTO TWO BINOMIAL FACTORS
WHERE THE TERMS IN THE FIRST POSITIONS
WILL BE THE FACTORS OF X SQUARED WHICH WILL BE X AND X.
AND THEN THE TERMS IN THE SECOND POSITIONS
WILL BE THE FACTORS OF NEGATIVE 27 THAT ADD TO NEGATIVE 6.
LET'S GO AHEAD AND LIST THE FACTORS OF NEGATIVE 27.
WE COULD HAVE NEGATIVE 3 x 9, NEGATIVE 9 x 3, NEGATIVE 1 x 27,
AND NEGATIVE 27 x 1.
OF COURSE IF WE CAN VISUALIZE THESE IN OUR MIND,
WE DON'T HAVE TO LIST THEM.
BUT NOW WE WANT THE TWO FACTORS THAT HAVE A SUM OF NEGATIVE 6
WHICH WOULD BE NEGATIVE 9 AND POSITIVE 3
WHICH MEANS ONE FACTOR WILL BE X - 9
AND ONE FACTOR WILL BE X + 3.
AND NOW THAT IT'S IN FACTORED FORM
AND WE HAVE OUR PRODUCT EQUAL TO 0,
EITHER THE FIRST FACTOR OF X - 9
MUST EQUAL 0 OR THE FACTOR OF X + 3 MUST EQUAL 0.
AND NOW, WE'LL SOLVE THESE TWO EQUATIONS FOR X
TO DETERMINE OUR SOLUTIONS.
SO HERE WE WOULD ADD 9 TO BOTH SIDES.
WE'D HAVE X = 9 OR HERE WE WOULD SUBTRACT 3 ON BOTH SIDES,
SO WE'D HAVE X = NEGATIVE 3.
AGAIN, THESE ARE THE TWO SOLUTIONS
TO OUR QUADRATIC EQUATION.
OKAY, WE'LL LOOK AT SOME MORE EXAMPLES IN THE NEXT VIDEO.