# Solving Quadratic Equations by Factoring 5 2 Solving Quadratic Equations by mikesanye

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```									 5.2: Solving Quadratic Equations
by Factoring
(p. 256)

Tucker Method of Factoring
Factoring Simple Trinomials

   Factoring simple polynomials in
the form of x2 + bx + c.          x2 + 5x + 6   Factors of:
+6
   An example would be:
◦ x2 + 5x + 6                      x     x       3   ●     2
-3   ●    -2
◦ First factor the last                          6   ●     1
number , +6 (include                          -6   ●    -1

the sign).
◦ Next, write the first
term (x2) without the
exponent on the top
of the bar.
Factoring Simple Trinomials

   Factoring x2 + 5x + 6              x2 + 5x + 6   Factors of:
◦ First factor the last number ,                    +6
+6 (include the sign).
x     x
◦ Find the factors of that                         3 + 2
-3 + -2
= +5
= -5
number (+6), that the            +3    +2       6 + 1
-6 + -1
= +7
= -7
sum (+) results in the
middle number. (+5).
◦ Place those factors (+3
and +2) in the box
under the bar.
Factoring Simple Trinomials

   Factoring x2 + 5x + 6          x2 + 5x + 6
◦ Now take the left side
Factors of:
+6
(x + 3) and place that
binomial in a parenthesis.    x     x          3 + 2 = +5
6 + 1 = +7
◦ Then take the right side     +3    +2
(x + 2) and place that
binomial in a
parenthesis.                         (x + 3) (x + 2) = x2 + 5x + 6

◦ You have now factored
the polynomial.
Factoring Simple Trinomials

   Now, Factor: x2 - 5x - 6
x2 - 5x - 6
◦ First factor the last                   Factors of:
number , -6 (include                       -6
the sign).                  x     x     -3 +    2    = -1
◦ Find the factors of                      3 +    -2   = +1
that number (-6),         -6    +1       6 +    -1   = +5
that the difference (-)                 -6 +     1   = -5

results in the middle
number. (-5).
◦ Place those factors (-
6 and +1) in the box
under the bar.
Factoring Simple Trinomials

   Factoring x2 - 5x - 6      x2 - 5x - 6
◦ Now take the left
Factors of:
-6
side (x - 6) and place
that binomial in a         x       x        -3   +     2   = -1

parenthesis.             -6       +1
3
6
+
+
-2
-1
= +1
= +5
◦ Then take the right                         -6   +     1   = -5

side (x + 1) and place
that binomial in a
parenthesis.                   (x - 6) (x + 1) = x2 - 5x - 6
◦ You have now
factored the
polynomial.
Factoring Simple Trinomials

   Factor: x2 - 7x + 12
x2 - 7x +12
◦ First factor the last                  Factors of:
number , +12                             +12
(include the sign).       x     x       2 + 6 = +8
◦ Find the factors of                    -2 + -6 = -8
that number (+12),       -3    -4       3 + 4 = +7
that the sum (+)                       -3 + -4 = -7
results in the middle                  12 + 1 = +13
number. (-7).                          -6 + 1 = -5

◦ Place those factors (-
3 and -4) in the box
under the bar.
Factoring Simple Trinomials

   Factor: x2 - 7x + 12       x2 - 7x + 12
◦ Now take the left
Factors of:
+12
side       (x - 3) and
place that binomial       x      x       2 + 6 = +8
in a parenthesis.                       -2 + -6 = -8
3 + 4 = +7
◦ Then take the right       -3    -4       -3 + -4 = -7
side (x - 4) and place                   12 + 1 = +13
-6 + 1 = -5
that binomial in a
parenthesis.              (x - 3) (x - 4) = x2 - 7x + 12
◦ You have now
factored the
polynomial.
Factoring Trinomials
ax2 + bx + c

 Factoring trinomials in the
5x2 - x -18
form of: ax2 + bx + c.                            Factors of:
5 ● -18 = -90
 An example would be:
5x    5x
◦ 5x2 - x –18                                              1 ●
2 ●
-90
-45
◦ First, place the first term on                            3 ●   -20
5 ●   -18
both sides of the bars.                                   6 ●   -15
9 ●
(without the exponent)                                   10 ●
-10
-9
◦ Second, multiply the first                               Etc……
(5) and the last number
(-18).
◦ Find the factors of -90.
Factoring Trinomials
ax2 + bx + c

   Factoring trinomials in the
5x2 - x -18
form of: ax2 + bx + c.                       Factors of:
-90
 Factor: 5x2    - x –18
5x    5x     1 + -90 = -89
◦ After finding the factors,                  2 + -45 = - 44
choose the set whose            9    -10    3 + -20 = -17
5 + -18 = - 13
difference equals the                       6 + -15 = -9
middle number, -1.                          9 + -10 = -1
(Remember that –x is the                   10 + -9 = +1
same as –1x).                              Etc……

◦ Place that set with signs
under the bar.
Factoring Trinomials
ax2 + bx + c

   Factoring trinomials in the
5x2 - x -18
form of: ax2 + bx + c.                         Factors of:
-90
   To finish factoring you
5x   5x
must reduce each side of                        1 + -90 = -89
2 + -45 = - 44
the bar if possible.             9     -10      3 + -20 = -17
5 + -18 = - 13
   The 5x over –10 will                  1x        6 + -15 = -9

2
9 + -10 = -1
reduce to 1x (or x) over -                     10 + -9 = +1
2.                                             Etc……
(5x + 9) (x - 2) = 5x2 - x –18
   Now, move each side into
Factoring Trinomials
ax2 + bx + c

   Another example would
be:                        9x2 + 18x -16   Factors of:
9 ● -16 = -144
◦ 9x2 +18x – 16
◦ First, place the first       9x    9x              12 ● -12
4 ● -38
term on both sides of                              3 ● -48
the bars. (without the                             6 ● -24
-6 ● 24
exponent)                                         -4 ● 36
-2 ● 72
◦ Second, multiply the                              Etc……
first (9) and the last
number (-16).
◦ Find the factors.
Factoring Trinomials
ax2 + bx + c

   Factoring trinomials in
9x2 + 18x -16
the form of: ax2 + bx + c.                   Factors of:
-144
   Factor: 9x2 + 18x –16            9x    9x    12 + -12 = 0
◦ After finding the                           4 + -38 = -34

factors, choose the            -6   +24     3 + -48 = -45
6 + -24 = -18
set whose difference                       -6 + 24 = +18
-4 + 36 = +32
equals the middle                          -2 + 72 = +70
Etc……
number, +18.
◦ Place that set with
signs under the bar.
Factoring Trinomials
ax2 + bx + c

   Factor: 9x2 + 18x –16
9x2 + 18x -16
   To finish factoring you                      Factors of:
-144
must reduce each side of
the bar if possible.           9x    9x       12 + -12 = 0
4 + -38 = -34
   The 9x over –6 will            -6    +24       3 + -48 = -45
6 + -24 = -18
reduce to 3x over -2.           3x     3x     -6 + 24 = +18
-4 + 38 = +34
   The 9x over +24 will            -2     +8     -2 + 72 = +70
Etc……
reduce to 3x over +8.
   Now, move each side into    (3x -2) (3x +8) = 9x2 +18x –16
Factoring Trinomials
ax2 + bx + c

   The last example contains a
‘Greatest Common Factor’:       2t2 + 3t - 5
◦ 6t3 + 9t2 – 15t
◦ First factor out the
Greatest Common Factor
(GCF) which is 3t.
◦ Go ahead and place the
do not forget it.
◦ Now, place the remaining       3t(   )(     ) = 6t3 + 9t2 – 15t
the bars.
Factoring Trinomials
ax2 + bx + c

   The last example contains a
‘Greatest Common Factor:            2t2 + 3t - 5                Factors of:
◦ 6t3 + 9t2 – 15t = 3t(2t2 +3t –                                  -10 = 2 ● -5
5)
2t       2t               1 + -10 = -9
◦ Next, place the first term                                     2 + -5 = -3
on both sides of the bars.               5        -2           5 + -2 = +3
(without the exponent)                                         10 + -1 = +9
◦ Then, multiply the first
number (2) and the last
number (-5).
◦ Find the factors that the
3t(       )(        ) = 6t3 + 9t2 – 15t
difference (-) will give you
the middle term (+3). Place
those factors under the bar.
Factoring Trinomials
ax2 + bx + c

   To finish factoring you     2t2 + 3t - 5
must reduce each side of                          Factors of:
-10 = 2 ● -5
the bar if possible.
2t 2t             1 + -10 = -9
   The left side of the bar                            2 + -5 = -3
5    -2
does not reduce, but the                            5 + -2 = +3
10 + -1 = +9
right side does reduce to              1t
-1
1t (or t) over –1.
   Now, move each side into
3t(2t +5)(t - 1) = 6t3 + 9t2 – 15t
Factoring Trinomials
Tucker Method
Factor these trinomials completely:
1. 6y2 + 13y + 6
2. 15t2 + 19t – 10
3. 18x2 – 24x + 6
4. 12x2 – 28x – 24
5. 15x3 + 19x2 – 10x
Zero Product Property

 Let A and B be real numbers or algebraic
expressions. If AB=0, then A=0 or B=0.
 This means that If the product of 2 factors
is zero, then at least one of the 2 factors
 Ex: If xy=0, then either x=0 or y=0.
Example: Solve.
2t2-17t+45=3t-5

2t2-17t+45=3t-5    Set eqn. =0
2t2-20t+50=0       factor out GCF of 2
2(t2-10t+25)=0     divide by 2
t2-10t+25=0        factor left side
(t-5)2=0           set factors =0
t-5=0              solve for t
+5 +5
Example: Solve.
x2+3x-18=0

x2+3x-18=0          Factor the left side
(x+6)(x-3)=0        set each factor =0
x+6=0 OR x-3=0      solve each eqn.
-6 -6      +3 +3
x=-6 OR x=3       check your solutions!
Example: Solve.
3x-6=x2-10

3x-6=x2-10           Set = 0
0=x2-3x-4            Factor the right side
0=(x-4)(x+1)         Set each factor =0
x-4=0 OR x+1=0       Solve each eqn.
+4 +4       -1 -1
x=4 OR x=-1          Check your solutions!
Finding the Zeros of an Equation
   The Zeros of an equation are the x-
intercepts !

 First, change y to a zero.
 Now, solve for x.
 The solutions will be the zeros of the
equation.
Example: Find the Zeros of
y=x2-x-6
y=x2-x-6                Change y to 0
0=x2-x-6                Factor the right side
0=(x-3)(x+2)            Set factors =0
x-3=0 OR x+2=0          Solve each equation
+3 +3      -2 -2
x=3 OR x=-2            Check your solutions!

If you were to graph the eqn., the graph would
cross the x-axis at (-2,0) and (3,0).
Assignment

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