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Chapter 9

Quadratic Equations.

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In this chapter we will look at…

9.1 Square Roots 9.2 Solving Quadratic Equations

– By finding square roots

9.3 Simplifying Radicals 9.4 Graphing Quadratic Functions 9.5 Solving Quadratics by Graphing 9.6 Solving Quadratics using the Quadratic Formula 9.7 The Discriminant 9.8 Graphing Quadratic Inequalities

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Watch this…

Write 5 important ideas from the clip.

We will be sharing these in class.

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Square Roots

9.1 How many squares are on each side of a chess board?

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Square Roots All positive real numbers have 2 square roots.

one positive square rootone negative square root

Square Roots are written using a radical sign.

√ The Number inside the radical sign is the radicand.

Square Roots Undo Numbers Squared. 32 = 3•3= 9 therefore V9 = 3 the positive square root. √9 also = -3 the negative square root, because (-3)(-3) = 9 ± √9 means the positive and negative square roots of 9.

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Perfect Squares Recall Irrational Numbers include the

square root of any non perfect square.

Perfect Square are rational numbers.– They are integers.

The square of an integer is a perfect square.

√n √1 √4 √9 √16 √25 √36 √49 √64 √81 √100 √121 √144

±n ±1 ±2 ±3 ±4 ±5 ±6 ±7 ±8 ±9 ±10 ±11 ±12

12=1 22=4 32=9 42=16 52=25 62=3672

_=49

82=64 92=81102=1

00112=1

21122=1

44

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The square root of a non-perfect square.

√n is an irrational number.

An Irrational number is a number

that can not be written as a

quotient of integers aka

fraction. √2 = 1.414, this is a non-perfect square,

it is irrational.

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Radical Expressions A radical expression is any expression

written with a radical. The radical acts as grouping

symbol. What is under the radical

is simplified first.

Then take the root of what is underneath.

Evaluating a radical expression is the same as evaluating any expression

Write, replace and simplify.

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Evaluate the radical…

√(b2 - 4ac)• When a =1 , b = -2 and c = -3

Replace the variable with its value and evaluate.

√((-2)2 - 4(1)(-3)) = √4 +12 = √16 = 4

QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.

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Square roots and TI

You can evaluate a square root on your calculator and

round the result if asked to do so.

Here the + - means you have to evaluate twice, one

for the positive square root and once for the

negative square root. Our expression has two solutions:

€

1± 2 3

4

QuickTime™ and aPhoto - JPEG decompressorare needed to see this picture.

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Evaluate the Radical

Remember the + - means you must evaluate twice.

Once for the positive square root Once for the negative square root. 1.24, 3.76 8.82, 3.17 -2.90, .57

€

2 ± 3

3 6 ± 8

7 ± 3 12

−6

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questionsSummary:

All positive real numbers have TWO square roots.

– One positive– One negative

– Written with a plus minus sign ±

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Simplifying Roots

Taking the square root undoes a square.• √16 = ±4 because 42 = 16

√ of a perfect square is rational, an integer.

• x2 = 81 the x = 9 and x = -9 or x = ±9

√ of non perfect square is irrational, written as a radical expression.

•y2 = 3 then y = ±√3

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Solving Quadratic Equations

by finding square roots.

9.2

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The quadratic equation… ax2 + bx + c = 0

– In standard form– The leading coefficient a ≠ 0

When b = 0 the equation becomes

ax2 + c = 0 Here you can solve for x by isolating the

squared variable and using the square root. √

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When ax2 + c = 0

x2 - 4 = 0

n2 - 49 = 0

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The square root of a non perfect square

When the answer to a squared number is

not a perfect square,

n2 = 5 The answer will be given as a

radical expression– more exact answer then using a decimal.

Here: n = √5 AND -√5 Or n = ±√5

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Solutions and the Quadratic

NO Real Solution No real solution if

the square of a number equals is negative.

y2 = -1 Here the square of a real

number is NEVER negative!

There is no real solution when:

x2 = d and d < 0

ONE Real Solution

x2 = 0 The √0 = 0 There is no other

solution, zero is neither positive or negative.

ONE real solution when

x2 = d and d = 0

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Solutions and the Quadratic

TWO Real Solutions All positive real numbers have 2

square roots. one positive square root one negative square root

x2 - 81 = 0 x2 = 81 x = ± 9

There are 2 real solutions when:

x2 = d and d > 0

Determine the solutions by solving the quadratic.

x2 = 81 y2 + 11 = 0 5n2 - 25 = 0 2x2 - 2= 0

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Quadratics and Real Life When a ball (or any object) is dropped the speed at

which it falls continuously increases. Ignoring air resistance, the height of the ball h can be approximated by the falling object model:

h = -16t2 + s h is the height t is the time in seconds s is the initial height from where the ball is dropped.

This model lets you look at the height of the object at anytime during its fall.

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Falling Object Model Cat P’s most famous science teacher Mr.

Tschachler is a contestant in an egg dropping contest. The goal is to create a container for the egg so it can be dropped from a height of 32 feet without breaking.

Write a model for the egg’s height, disregard air resistance.

Use the model to determine how long it will

take the egg to reach the ground.

If h = -16t2 + s then…

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Algebraic Model for Mr. Tschachler’s egg drop. h = -16t2 + s What is the initial condition, where are we starting?

32 feet above ground. Replace the variable s with 32, this is the initial condition.

Ground level is represented by 0, replace the height h with 0.

The model for the egg drop is: 0 = -16t2 + 32 Solve for t, time.

0 = -16t2 + 32 -32 = -16t2

2 = t2

√2 = √t2

±√2 As a decimal, t ≈ 1.4 seconds

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questions??

Assignment

9.1 #25, 29, 31, 41, 45, 61, 62, 69,70, 80, 82

9.2 #24, 28, 33, 35, 42, 44, 59, 60