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Chapter 16
The Quadratic Formula
Algebra 1
Mr. Swartz
Proving the Quadratic Formula
Solve by completing the square
02 cbxax
a
cx
a
bx 2
2
22
42 a
b
a
b
a
c
a
b
a
bx
a
bx
2
2
2
22
44
a
c
a
b
a
bx
2
22
42
2
22
4
4
2 a
acb
a
bx
a
acb
a
bx
2
4
2
2
22
22
4
4
42 a
ac
a
b
a
bx
24LCM a
a
acbbx
2
42
a
acb
a
bx
2
4
2
2
a2
Quadratic Formula
There was a negative boy who couldn’t decide to
go to this radical party. Because the boy was
square, he lost out on 4 awesome chicks so he
cried his way home when it was all over at 2 AM.
b 2b ac4x
VIDEO
Quadratic Formula
a
acbbx
2
42
02 cbxax
IF
THEN
#1 Solve using the quadratic formula.
0273 2 xx
a
acbbx
2
42
2 ,7 ,3 cba
)3(2
)2)(3(4)7()7( 2 x
6
24497 x
6
57 x
6
12x
6
2x
3
1 ,2x
6
257 x
#2 Solve using the quadratic formula542 2 xx
a
acbbx
2
42
5 ,4 ,2 cba
)2(2
)5)(2(4)4()4( 2 x
4
40164 x
4
564x
4
1424 x
0542 2 xx
2
141x
9.2x 9.0x
4
1444 x
Solve using the quadratic formula
3.
4.
mm 1083 2
xx 16642
#4 Solve using the quadratic formula
mm 1083 2
a
acbbm
2
42
8 ,10 ,3 cba
)3(2
)8)(3(4)10()10( 2 m
6
9610010 m
6
1410m
6
24m
6
4m
3
2 ,4 m
08103 2 mm
6
19610 m
#5 Solve using the quadratic formula
xx 16642
a
acbbx
2
42
64 ,16 ,1 cba
)1(2
)64)(1(416162
x
2
25625616 x
2
016 x
8x
064162 xx
Solve 11n2 – 9n = 1 by the quadratic
formula.
11n2 – 9n – 1 = 0, so
a = 11, b = -9, c = -1
)11(2
)1)(11(4)9(9 2
n
22
44819
22
1259
22
559
The Quadratic Formula
Example
)1(2
)20)(1(4)8(8 2
x
2
80648
2
1448
2
128 20 4 or , 10 or 2
2 2
x2 + 8x – 20 = 0 (multiply both sides by 8)
a = 1, b = 8, c = 20
8
1
2
5Solve x2 + x – = 0 by the quadratic formula.
The Quadratic Formula
Example
Solve x(x + 6) = 30 by the quadratic
formula.
x2 + 6x + 30 = 0
a = 1, b = 6, c = 30
)1(2
)30)(1(4)6(6 2
x
2
120366
2
846
So there is no real solution.
The Quadratic Formula
Example
Solve 12x = 4x2 + 4.
0 = 4x2 – 12x + 4
0 = 4(x2 – 3x + 1)
Let a = 1, b = -3, c = 1
)1(2
)1)(1(4)3(3 2
x
2
493
2
53
Solving Equations
Example
GCF is 4
By factoring out a GCF it helps
by making your a, b, and c smaller
Solve the following quadratic equation.
02
1
8
5 2 mm
0485 2 mm
0)2)(25( mm
02025 mm or
25
2 mm or
Solving Equations
Example
ELIMINATE FRACTIONS
Multiply by the GCF, 8
Wait, it factors, a*c
5* -4 = -20 Factors that add
To +8 are +10, -2.
*You can solve by the Quadratic
Formula if you prefer*
Solving Quadratic Equations
Steps in Solving Quadratic Equations
1) If the equation is in the form (ax+b)2 = c, use
the square root property to solve.
2) If not solved in step 1, write the equation in
standard form.
3) Try to solve by factoring.
4) If you haven’t solved it yet, use the quadratic
formula.
The Discriminant
Discriminant In the quadratic
formula, the expression
underneath the radical
that describes the
nature of the roots.a
acbbx
2
42
acb 4 nt discrimina 2
Understanding the discriminant
Discriminantacb 42
# of real roots
042 acb 2 real roots
042 acb 1 real roots
042 acb No real roots
#6 Using the discriminant
0134 2 yy
acb 4 nt discrimina 2
1 ,3 ,4 cba
)1)(4(4)3( nt discrimina 2
169 nt discrimina
52 nt discrimina
052
roots real 2
#7 Using the discriminant
54 2 xx
acb 4 nt discrimina 2
5 ,1 ,4 cba
)5)(4(4)1( nt discrimina 2
801 nt discrimina
79 nt discrimina
079
roots real 0
Using the discriminant
8.
9.
542 2 xx
484 2 xx
56nt discrimina
roots real 2
0 nt discrimina
root real 1
#8 Using the discriminant
542 2 xx
acb 4 nt discrimina 2
0542 2 xx
)5)(2(4)4( nt discrimina 2
4061 nt discrimina
56nt discrimina
056
roots real 2
#9 Using the discriminant
484 2 xx
acb 4 nt discrimina 2
0484 2 xx
)4)(4(4)8( nt discrimina 2
6446 nt discrimina
0 nt discrimina
00
root real 1
