Chapter 5 - Test Review
Go to this website for extra help:
go.hrw.com/hrw.nd/gopages
Look for parent resources in the 2007 version of our book. Chapter 5 part a and chapter 5 part b.
Simplifying Radicals Video (How the heck do you simplify a square root?)
http://video.google.com/videosearch?q=Algebra+2+-+Trigonometry+Intro&hl=en&emb=0&aq=f&aq=-1&oq=#q=Simplifying+radicals&hl=en&emb=0
Don't forget to check out the older post on chapter 5 review and review sheets!
Monday, April 20, 2009
Wednesday, April 15, 2009
GTA3 3.2 Homework Review Notes
42. a^2 - b^2
(1 - sin^2 ?)(1 + sin^2 ?)
option a -looking at the factor as a difference squares
1^2 - (sin ^2 ?)^2
1 - sin ^4 ?
option b -using fundamental identity
cos^2 ?(1+sin^2 ?)
cos^2 ? + cos^2 ? sin^2 ?
option c -looking at the first binomial as a difference of squares
1^2 - Sin^2 ?
(1 - sin ?)(1 + sin ?)(1 + sin^2 ?)
let's try starting with the more complicated side!
2 cos^2 ? - cos ^4 ? factor out a cos ^2 ?
cos^2 ? (2- cos^2 ?) use fundamenal identity
1 - sin ^2 ? (2 - (1 - sin^2 ?)) oh oh, now we are rollin! distibute the minus in the second parenthesis
(1 - sin2 ?)( 2-1+sin^2 ?)
(1 - sin2 ?)(1+sin^2 ?)
(1 - sin^2 ?)(1 + sin^2 ?)
option a -looking at the factor as a difference squares
1^2 - (sin ^2 ?)^2
1 - sin ^4 ?
option b -using fundamental identity
cos^2 ?(1+sin^2 ?)
cos^2 ? + cos^2 ? sin^2 ?
option c -looking at the first binomial as a difference of squares
1^2 - Sin^2 ?
(1 - sin ?)(1 + sin ?)(1 + sin^2 ?)
let's try starting with the more complicated side!
2 cos^2 ? - cos ^4 ? factor out a cos ^2 ?
cos^2 ? (2- cos^2 ?) use fundamenal identity
1 - sin ^2 ? (2 - (1 - sin^2 ?)) oh oh, now we are rollin! distibute the minus in the second parenthesis
(1 - sin2 ?)( 2-1+sin^2 ?)
(1 - sin2 ?)(1+sin^2 ?)
5.6 Quadratic Equations and Complex N...
5.6 Quadratic Equations and Complex Numbers
(by: nate epstine)
Discriminant= (b^2-4ac)- used to determine the number of REAL, UNIQUE solutions a quadratic equation has
If (b^2-4ac)>0, there are 2 distinct, real solutions
If (b^2-4ac)=0, there is 1 real solution (a double root)
If (b^2-4ac)<0, there are 0 real solutions
Example 1:
Find the Discriminant of each equation, then determine the number of real solutions for each equation
A) 2x^2+4x+1=0 B) 2x^2+4x+2=0 C) 2x^2+4x+3=0
4^2-4(2)(1) 4^2-4(2)(2) 4^2-4(2)(3)
=8 =0 =(-8)
Imaginary Number
Imaginary Number= i is defined to be the square root of -1
i^1 i^2 i^3 i^4
---------------------------
i -1 -i 1
Example 4-> Use the Quadratic Formula to solve
3x^2-7x+5=0
x=(7±square root of (-7^2-4(3)(5))/2(3)
7±square root of (-11)/6
QUIZ THURSDAY→ 5.4-5.6
___________________________________________________________________
Addition to Nate's Notes:
Complex Numbers: Any number that can be written as a+bi
a - real part
b - imaginary part
Notes Continued:(by: Halina A. Wiktor)
Division: We don't really divide, we multiply by the Complex Conjugate. This is called factoring the denominator.
Complex Conjugate: The complex conjugate of a +bi is denoted a+bi, is a-bi. All you do to find the conjugate is change the operator between the real and imaginary parts.
Ex. 9: Find -2-3i -> -2+3i
Ex. 10: Simplify 2+5i/2-3i (Write your answer in standard form)
Step 1: multiply the numerator and the denominator by the complex conjugate of the denominator (2+5i)/(2-3i) (2+3i)/(2+3i)
Step 2: Foil and simplify 4+6i+10+15i^2/4+6+6i-9i^2 = -11+16i/13
Step 3: Write the answer in standard form: -11/13 + 16i/13
Graphing Complex Numbers: Complex numbers are graphed in the complex plane.
Complex Plane: The x-axis is the real axis and the y-axis is the imaginary axis.
Ex. 11: Graph 2-3 in the complex plane.
Magnitude of a Complex Number: denoted by |a+bi|, means: find the distance the complex is from the origin. Absolute value, means distance from zero. To do this:
|a+bi| = (square root of) a^2 +b^2
Ex. 12: Evaluate |-2-3i| then sketch.
|-2-3i| = (square root of) (-2)^2 + (-3)^2 = (square root of) 4+9 = (square root of) 13
Ex. 13: Which value is greater |2+4i|or |1-5i|? Prove algebraically
|2+4i|= (square root of) 2^2+ 4^2 + (square root of) 4+16 = (square root of) 20
|1-5| = (square root of) 1+25 = (square root of) 26 -> greater
(by: nate epstine)
Discriminant= (b^2-4ac)- used to determine the number of REAL, UNIQUE solutions a quadratic equation has
If (b^2-4ac)>0, there are 2 distinct, real solutions
If (b^2-4ac)=0, there is 1 real solution (a double root)
If (b^2-4ac)<0, there are 0 real solutions
Example 1:
Find the Discriminant of each equation, then determine the number of real solutions for each equation
A) 2x^2+4x+1=0 B) 2x^2+4x+2=0 C) 2x^2+4x+3=0
4^2-4(2)(1) 4^2-4(2)(2) 4^2-4(2)(3)
=8 =0 =(-8)
Imaginary Number
Imaginary Number= i is defined to be the square root of -1
i^1 i^2 i^3 i^4
---------------------------
i -1 -i 1
Example 4-> Use the Quadratic Formula to solve
3x^2-7x+5=0
x=(7±square root of (-7^2-4(3)(5))/2(3)
7±square root of (-11)/6
QUIZ THURSDAY→ 5.4-5.6
___________________________________________________________________
Addition to Nate's Notes:
Complex Numbers: Any number that can be written as a+bi
a - real part
b - imaginary part
Notes Continued:(by: Halina A. Wiktor)
Division: We don't really divide, we multiply by the Complex Conjugate. This is called factoring the denominator.
Complex Conjugate: The complex conjugate of a +bi is denoted a+bi, is a-bi. All you do to find the conjugate is change the operator between the real and imaginary parts.
Ex. 9: Find -2-3i -> -2+3i
Ex. 10: Simplify 2+5i/2-3i (Write your answer in standard form)
Step 1: multiply the numerator and the denominator by the complex conjugate of the denominator (2+5i)/(2-3i) (2+3i)/(2+3i)
Step 2: Foil and simplify 4+6i+10+15i^2/4+6+6i-9i^2 = -11+16i/13
Step 3: Write the answer in standard form: -11/13 + 16i/13
Graphing Complex Numbers: Complex numbers are graphed in the complex plane.
Complex Plane: The x-axis is the real axis and the y-axis is the imaginary axis.
Ex. 11: Graph 2-3 in the complex plane.
- Go over 1 on the real-axis and down 3 on the imaginary axis
- Put a dot there
- Draw a dotted line from the origin to that dot. (This is how i want complex numbers graphed)
Magnitude of a Complex Number: denoted by |a+bi|, means: find the distance the complex is from the origin. Absolute value, means distance from zero. To do this:
|a+bi| = (square root of) a^2 +b^2
Ex. 12: Evaluate |-2-3i| then sketch.
|-2-3i| = (square root of) (-2)^2 + (-3)^2 = (square root of) 4+9 = (square root of) 13
Ex. 13: Which value is greater |2+4i|or |1-5i|? Prove algebraically
|2+4i|= (square root of) 2^2+ 4^2 + (square root of) 4+16 = (square root of) 20
|1-5| = (square root of) 1+25 = (square root of) 26 -> greater
Thursday, April 2, 2009
Wednesday, April 1, 2009
GTA3 3.1 Homework Review
This has been posted to the googledoc titled 3.1 Basic Identities, make notes or ask questions. I have included the work below as well.
sin x = + 1/sqrt (1+cot^2 x)
Don't forget to add notes to google docs on 3.2 reading!
3.1 Homework Review
13.) sin x in terms of cot x
sin x = 1/csc x
(sin^2 x) = (1/csc x)^2
sin^2 x = 1/(csc^2 x)
sin^2 x = 1/(1+cot^2 x)
sqrt (sin^2 x) = + sqrt (1/(1+cot^2 x))
13.) sin x in terms of cot x
sin x = 1/csc x
(sin^2 x) = (1/csc x)^2
sin^2 x = 1/(csc^2 x)
sin^2 x = 1/(1+cot^2 x)
sqrt (sin^2 x) = + sqrt (1/(1+cot^2 x))
Don't forget to add notes to google docs on 3.2 reading!
Subscribe to:
Comments (Atom)