Square Roots and Solving

Quadratics with Square RootsReview 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 SquaresThe 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 = –749 = 7

100 = 10100 = –10

A. 49

B. 100

C. 225

15 is a square root, since 15 • 15 = 225.225 = 15225 = –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 = 12144 = –12

Find the two square roots of each number.

B. 144

C. 289

289 = 17289 = –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 = 4x = ±2

€

x 2 = 5

x = 5

€

x 2 = 0x = 0

€

x 2 = −1NO SOLUTION

SQUARE ROOT BOTH SIDES

SolveSolve 3x2 – 48 = 0

+48 +48

3x2 = 483 3x2 = 16

€

x 2 = 16x = ±4

Example 1: Solve the equation:1.) x2 – 7 = 9 2.) z2 + 13

= 5 +7 + 7

x2 = 16

€

x 2 = 16x = ±4

- 13 - 13

z2 = -8

€

z2 = −8NO SOLUTION

Example 2: Solve 9m2 = 169

9 9

m2 =

€

m2 = 1699

x = 1699

€

1699

Example 3: Solve 2x2 + 5 = 15

-5 -5

2x2 = 102 2

x2 = 5

€

x 2 = 5

x = 5

Example:

1. 2. 1083 2 x 1255 2 x

3 3

x2 = 36

€

x 2 = 36x = ±6

5 5

x2 = 25

€

x 2 = 25x = ±5

Example:3. 4264 2 x

+6 +6

4x2 = 484 4x2 = 12

€

x 2 = 12

x = 12

Examples:

4. 5. 953 2 x 2154

2

x

-3 -3

-5x2 = -12

-5 -5x2 = 12/5

€

x 2 = 125

x = 125

+5 +5

4 4

x2 = 104

€

x 2 = 104

x = 104€

x 2

4= 26