Section 4.3 Solving Quadratics by Factoring and the Square ...
Square Roots and Solving Quadratics with Square Roots
description
Transcript of Square Roots and Solving Quadratics with Square Roots
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