square roots and solving quadratics with square roots review 9.1-9.2
TRANSCRIPT
Square Roots and Solving
Quadratics with Square Roots
Review 9.1-9.2
GET YOUR COMMUNICATORS!!!!
Warm UpSimplify.
25 64
144 225
400
1. 52 2. 82
3. 122 4. 152
5. 202
Perfect SquareA number that is the
square of a whole numberCan be represented by
arranging objects in a square.
Perfect Squares
Perfect Squares
1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 =
16
Perfect Squares
1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16Activity: Calculate the perfect squares up to 152…
Perfect Squares
1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 5 x 5 = 25 6 x 6 = 36 7 x 7 = 49 8 x 8 = 64
9 x 9 = 81 10 x 10 =
100 11 x 11 =
121 12 x 12 =
144 13 x 13 =
169 14 x 14 =
196 15 x 15 =
225
Activity:Identify the following
numbers as perfect squares or not.
i. 16ii. 15
iii. 146iv. 300v. 324vi. 729
Activity:Identify the following
numbers as perfect squares or not.
i. 16 = 4 x 4ii. 15
iii. 146iv. 300
v. 324 = 18 x 18vi. 729 = 27 x 27
Perfect Squares: Numbers whose square roots are integers or quotients of
integers.
1316912144
1112110100
981864749
636525416
392411
Perfect SquaresOne property of a
perfect square is that it can be represented by a
square array. Each small square in the
array shown has a side length of 1cm.
The large square has a side length of 4 cm.
4cm
4cm 16 cm2
Perfect Squares
The large square has an area of 4cm x 4cm = 16
cm2.
The number 4 is called the square root of 16.
We write: 4 = 16
4cm
4cm 16 cm2
Square Root
A number which, when multiplied by itself, results
in another number.
Ex: 5 is the square root of 25.
5 = 25
Finding Square Roots
We can think “what” times “what” equals the larger
number.
36 = ___ x ___6 6
SO ±6 IS THE SQUARE ROOT OF 36
Is there another answer?
-6 -6
Finding Square Roots
We can think “what” times “what” equals the larger
number.
256 = ___ x ___16 16
SO ±16 IS THE SQUARE ROOT OF 256
Is there another answer?
-16 -16
Estimating Square Roots
25 = ?
Estimating Square Roots
25 = ±5
Estimating Square Roots
- 49 = ?
Estimating Square Roots
- 49 = -7
IF THERE IS A SIGN OUT FRONT OF THE RADICALTHAT IS THE SIGN WE USE!!
Estimating Square Roots
27 = ?
Estimating Square Roots
27 = ?
Since 27 is not a perfect square, we will leave it in a radical because that is an EXACT ANSWER.
If you put in your calculator it would give you 5.196, which is a decimal apporximation.
€
27
Estimating Square Roots
Not all numbers are perfect squares.
Not every number has an Integer for a square root.
We have to estimate square roots for numbers between
perfect squares.
Estimating Square Roots
To calculate the square root of a non-perfect square
1. Place the values of the adjacent perfect squares on a
number line.
2. Interpolate between the points to estimate to the nearest
tenth.
Estimating Square Roots
Example: 27
25 3530
What are the perfect squares on each side of 27?
36
Estimating Square Roots
Example: 27
25 3530
27
5 6half
Estimate 27 = 5.2
36
Estimating Square Roots
Example: 27
Estimate: 27 = 5.2
Check: (5.2) (5.2) = 27.04
Find the two square roots of each number.
7 is a square root, since 7 • 7 = 49.
–7 is also a square root, since –7 • –7 = 49.
10 is a square root, since 10 • 10 = 100.
–10 is also a square root, since –10 • –10 = 100.
49 = –7
49 = 7
100 = 10
100 = –10
A. 49
B. 100
C. 225
15 is a square root, since 15 • 15 = 225.225 = 15
225 = –15 –15 is also a square root, since –15 • –15 = 225.
A. 25
5 is a square root, since 5 • 5 = 25.–5 is also a square root, since –5 • –5 = 25.
12 is a square root, since 12 • 12 = 144.
–12 is also a square root, since –12 • –12 = 144.
25 = –525 = 5
144 = 12
144 = –12
Find the two square roots of each number.
B. 144
C. 289
289 = 17
289 = –17
17 is a square root, since 17 • 17 = 289.
–17 is also a square root, since –17 • –17 = 289.
Evaluate a Radical Expression
416124
)3(44)3)(1(4)2(4
.3,2,14
22
2
acb
candbawhenacbEvaluate
EXAMPLE SHOWN BELOW
Evaluate a Radical Expression
€
Evaluate b2 − 4ac when a = 3, b = −6, and c = 3.
€
b2 − 4ac = (−6)2 − 4(3)(3) = 36 − 4(9)
= 36 − 36 = 0 = 0
#1
Evaluate a Radical Expression
€
Evaluate b2 − 4ac when a = 5, b = 8, and c = 3.
€
b2 − 4ac = (8)2 − 4(5)(3) = 64 − 4(15)
= 64 −60 = 4 = ±2
#2
Evaluate a Radical Expression
€
Evaluate b2 − 4ac when a = −4, b = −9, and c = −5.
€
b2 − 4ac = (−9)2 − 4(−4)(−5) = 81− 4(20)
= 81−80 = 1 = ±1
#3
Evaluate a Radical Expression
€
Evaluate b2 − 4ac when a = −2, b = 9, and c = 5.
€
b2 − 4ac = (9)2 − 4(−2)(5) = 81− 4(−10)
= 81− (−40) = 121 = ±11
#4
SOLVING EQUATIONS
SOLVING MEANS “ISOLATE” THE VARIABLE
x = ??? y = ???
Solving quadratics
Solve each equation.a. x2 = 4 b. x2 = 5 c. x2 = 0 d. x2 = -1
€
x 2 = 4
x = ±2
€
x 2 = 5
x = 5
€
x 2 = 0
x = 0
€
x 2 = −1
NO SOLUTION
SQUARE ROOT BOTH SIDES
SolveSolve 3x2 – 48 = 0
+48 +48
3x2 = 483 3
x2 = 16
€
x 2 = 16
x = ±4
Example 1: Solve the equation:1.) x2 – 7 = 9 2.) z2 + 13
= 5 +7 + 7
x2 = 16
€
x 2 = 16
x = ±4
- 13 - 13
z2 = -8
€
z2 = −8
NO SOLUTION
Example 2:
Solve 9m2 = 1699 9
m2 =
€
m2 = 1699
x = 1699
€
169
9
Example 3:
Solve 2x2 + 5 = 15 -5 -5
2x2 = 10
2 2
x2 = 5
€
x 2 = 5
x = 5
Example:
1. 2. 1083 2 x 1255 2 x
3 3
x2 = 36
€
x 2 = 36
x = ±6
5 5
x2 = 25
€
x 2 = 25
x = ±5
Example:
3. 4264 2 x+6 +6
4x2 = 484 4
x2 = 12
€
x 2 = 12
x = 12
Examples:
4. 5. 953 2 x 2154
2
x
-3 -3
-5x2 = -12
-5 -5
x2 = 12/5
€
x 2 = 125
x = 125
+5 +5
4 4
x2 = 104
€
x 2 = 104
x = 104
€
x 2
4= 26