ALG 1B/ cdipaulo1 Chapter 9 Quadratic Equations.
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Transcript of ALG 1B/ cdipaulo1 Chapter 9 Quadratic Equations.
ALG 1B/ cdipaulo 1
Chapter 9
Quadratic Equations.
ALG 1B/ cdipaulo 2
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
ALG 1B/ cdipaulo 3
Watch this…
Write 5 important ideas from the clip.
We will be sharing these in class.
ALG 1B/ cdipaulo 4
Square Roots
9.1 How many squares are on each side of a chess board?
ALG 1B/ cdipaulo 5
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.
ALG 1B/ cdipaulo 6
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
ALG 1B/ cdipaulo 7
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.
ALG 1B/ cdipaulo 8
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.
ALG 1B/ cdipaulo 9
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.
ALG 1B/ cdipaulo 10
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.
ALG 1B/ cdipaulo 11
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
ALG 1B/ cdipaulo 12
questionsSummary:
All positive real numbers have TWO square roots.
– One positive– One negative
– Written with a plus minus sign ±
ALG 1B/ cdipaulo 13
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
ALG 1B/ cdipaulo 14
Solving Quadratic Equations
by finding square roots.
9.2
ALG 1B/ cdipaulo 15
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. √
ALG 1B/ cdipaulo 16
When ax2 + c = 0
x2 - 4 = 0
n2 - 49 = 0
ALG 1B/ cdipaulo 17
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
ALG 1B/ cdipaulo 18
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
ALG 1B/ cdipaulo 19
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
ALG 1B/ cdipaulo 20
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.
ALG 1B/ cdipaulo 21
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…
ALG 1B/ cdipaulo 22
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
ALG 1B/ cdipaulo 23
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