Math 096–Square Root Property–page 1
A.
Background
1.
We are going to talk about quadratic equations (2nd degree equations).
a.
Standard form:
ax2 + bx + c = 0
b.
Other forms:
ax2 + bx = 0
ax2 + c = 0
2.
Techniques for solving quadratic/2nd degree equations:
factoring
square root property
complete the square
quadratic formula:
3.
Types of solutions:
rational
irrational
complex
B.
Review
4.
x2 ! 64 = 0
Solve by factoring
(x + 8)(x ! 8) = 0
x + 8 = 0 or x ! 8 = 0
x = !8 or x = 8
However, you could also solve like this to get the same results:
x2 ! 64 = 0
Isolate x2
x2 = 64
Now square root both sides to go from x2 to x; use
on
constant because you still want two results
Now simplify the radical
x=±8
Notice you get the same answer whether you factored or whether you used the square root
property. When you see x = ± 8, that means x = 8 or !8.
5.
We will use the square root property to find results of problems we couldn’t solve before.
We’ll use the square root property on these type problems: ax2 + b = 0. Isolate the square
term first. Then square root both sides. Simplify the radical. Look at all the examples.
Math 096–Square Root Property–page 2
C.
Solve by the Square Root Property
6.
x2 ! 11 = 0
x2 = 11
Can’t factor (with integers) so isolate x2:
Square root both sides
x=
Since 11 doesn’t factor, you are finished!
x2 ! 20 = 0
x2 = 20
Isolate x2
Square root both sides
7.
Factor 20 to simplify: 20 = 2 @ 2 @ 5
x=
NOTE:
Suppose the equation read this:
x2 = 20
Notice x2 is already isolated so you can just square root both sides!
Then you can go ahead and simplify
x=
8.
x2 + 63 = 0
x2 = !63
x=
Think: 63 = 3 @ 3 @ 7
x=
9.
Sometimes you need to do more to isolate x2:
4x2 ! 20 = 140
4x2 = 140 + 20
4x2 = 160
x2 = 160 ÷ 4
x2 = 40
Add 20 to both sides
Now divide by 4 on both sides
Now square root both sides
Think: 40 = 2 @ 2 @ 2 @ 5
x=
Math 096–Square Root Property–page 3
D.
Extended Ideas
10.
Sometimes you will have (expression)2 = constant. You can still square root both sides to
remove the power of 2. Look at the examples.
11.
(x + 3)2 = 24
x+3=
24 = 2 @ 2 @ 2 @ 3
x+3=
Now get x alone; “move” 3 across equal sign
x=
12.
(x ! 7)2 = 25
25 = 5 @ 5
x!7=±5
x=7±5
x = 7 + 5 or 7 ! 5
x = 12 or 2
13.
Get x alone; “move” 7 across equal sign
Don’t stop – you need to simplify more!
Here’s an interesting one:
(2x + 3)2 = 98
98 = 2 @ 7 @ 7
2x + 3 =
2x =
Now divide by 2 on both sides:
x=
Homework. You try these.
1.
x2 ! 7 = 0
2.
x2 ! 12 = 0
3.
x2 ! 36 = 0
4.
x2 + 18 = 0
5.
x2 + 50 = 0
6.
x2 + 49 = 0
7.
x2 = 15
8.
x2 = 28
9.
x2 = !44
10.
2x2 + 8 = 56
11.
3x2 ! 18 = 45
12.
5x2 + 10 = !90
Math 096–Square Root Property–page 4
13.
(x + 3)2 = 7
14.
(x + 5)2 = 80
15.
(x + 1)2 = !48
16.
(x ! 2)2 = 25
17.
(x ! 6)2 = 100
18.
(x + 4)2 = 49
19.
(x + 7)2 = 36
20.
(x ! 9)2 = 81
21.
(x + 2)2 = !25
Answer Key.
1.
x=
2.
x=
3.
x=±6
4.
x=
5.
x=
6.
x=
7.
x=
8.
x=
9.
x=
10.
x=
11.
x=
12.
x=
13.
x=
14.
x=
15.
x=
16.
x=2±5
x = 2 + 5 or 2 ! 5
x = 7 or !3
17.
x = 6 ± 10
x = 6 + 10 or 6 ! 10
x = 16 or !4
18.
x = !4 ± 7
x = !4 + 7 or !4 ! 7
x = 3 or !11
19.
x = !7 ± 6
20.
x = !7 + 6 or !7 ! 6
x = !1 or !13
x=9±9
x = 9 + 9 or 9 ! 9
x = 18 or 0
21.
x=