COLLEGE ALGEBRA
Section 1.2
MATH 161
TEAGUE
Quadratic Equations
Example 1) FACTOR COMPLETELY 80 – 5x2
Factor the 5 out first 5(16-x2)
Then use difference of two squares 5(4-x)(4+x)
Example 2) (x+13)2 -7(x+13)
Factor out the common factor (x+13) to get (x+13)(x+13-7)
Then simplify
(x+13)(x+6)
Example 3) 28 (p2 –1) = 33p
Distribute 28p2 – 28 = 33p
Then move all to left to factor 28p2 – 33p – 28 = 0
Factor (7p + 4)(4p – 7)
Example 4) 42𝑥 − 13 =
42
𝑥
4 7
x=− ,
7 4
Multiply all by x
42x2 - 13x = 42
Then move all to left to factor 42p2 – 13p – 42 = 0
Factor (7p + 6)(6p – 7)
Example 5)
7(𝑥−3)
𝑥−4
2
−2
𝑥
𝑥(𝑥−4)
+ =
7x(x-3) + 2(x-4) = -2
6 7
x=− ,
7 6
Multiply all by x(x-4) to get
Then solve for 7x2-21x+2x-8=-2
7x2-19x-6=0
(7x+2)(x-3)
2
x = − ,3
7
Example 6) (2y+3)2 = 9 Take the square root of both sides 2y+3=3 and 2y+3= -3
2y+3= 3
y =0
2y+3= -3
y = −3
Example 7) Quadratic formula: 9y2-y+7 = 0
1±√1−4∙9∙7
18
=
1±√−251
18
Cannot have a negative under the radical
2
Example 8) Quadratic formula: 𝑥 2-x9
2x2-9x-1 = 0
x=
1
=0
9
9±√81−4∙2∙(−1)
4
4
,
𝑥−4
+
4
𝑥+1
4
9±√89
4
=0
𝑥
𝑥2
1±√1−4∙8∙(−4)
8x2-x-4 = 0
𝑥
=
4
1
Example 10)
Multiply by 9 first
9+√89 9−√89
Example 9) Quadratic formula: 8 − −
x=
NO SOLUTION
16
Multiply by x2 first
=
1±√129
16
1+√129 1−√129
=
16
,
19𝑥−1
𝑥 2 −3𝑥−4
(x-4)(x+1)
16
Multiply all by (x+1)(x-4) to get
Then solve for x2+x+4x-16=19x-1
x2-14x-15=0
(x-15)(x+1)
x = -1 and 15 however (-1-1)=0 is undefined so x = 15
x(x+1)+4(x-4) = 19x-1
Example 11) A rectangular garden has length of 9 feet greater than the width. Find
the length and width if the area is 400ft2?
Let l = w+9
w (w+9) = 400 w2 +9x – 400 = 0
(w+25) (w-16) then w = 16
width is 16ft and length is 25ft
Example 12) A ball is thrown vertically upward from a building 128 ft tall with
initial velocity of 112 ft/sec. The distance s (in feet) of the ball
from the ground after t seconds is s = 128 + 112t – 16t2
a) After how many seconds does the ball hit the ground?
-16t2 + 112t + 128=0
-16(t2 – 7t – 8) =0
(t-8) (t-11) =0
t = 8 seconds
b) After how many seconds does the ball pass the top of the building on its
way down?
-16t2 + 112t + 128=128
112t = 16t2
16t (t-7)=0
t = 7 seconds