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Algebra 2: Conics Preview
Transcript of Algebra 2: Conics Preview
Getting Ready to Tackle CONICS Objectives Prerequisites Standards Let's get Started! Each student will be able to complete the square of a binomial expression.
Each student will be able to solve problems by completing the square.
Each student will be able to explain how to complete the square using Algebra Tiles. What are pre-requisites? Prerequisites is all the stuff we assume you know! Anwer TRUE or FALSE to the following QUESTIONS. 1. I know how to complete the square .
2. I know how to expand (x + a)^2
3. I know how to find the equation of a line
given a point and slope
______________ How does completing the square looks in pictures? This.....YOU ARE RIGHT! Let's start with a simple problem
x^2 + 2x + _______
what goes in the blank? What should go here?
WE NEED TO ADD "ONE" TO
COMPLETE THE SQUARE So far we have
x^2 + 2x + 1
What if we have x^2 + 4x + __?
? IT'S FOUR!!!!!! Can you complete the square for
x^2 + 6x + _______? If we arrange 3x on each side of the square
we get......... So the answer is
x^2 + 6x + 9 Let's come up with a general rule
for completing the square x^2 + 2x + 1 x^2 + 4x + 4 x^2 + 6x + 9 IN EACH CASE, the coefficient from
the second term was divided into 2 equal parts,
and then, it was raised to the SECOND power..... x^2 + 2x + 1
2/2 = 1
1^2 = 1
CHECK! x^2 + 4x + 4
4/2 = 2
2^2 = 4
CHECK! x^2 + 6x + 9
6/2 = 3
3^2 = 9
CHECK! Write this down
When an expression on one side of a quadratic equation is not a perfect
square trinomial, we use the process called "completing the square" to obtain
a perfect square trinomial.
We generate a perfect square trinomial by adding (b/2)^2 to the expression
x^2 + bx.
EXAMPLES Solve x^2 + 4x = 5 We said before...... In this problem, b = 4
(b/2)^2 = (4/2)^2=2^2=4 x^2 + 4x + 4 = 5 + 4
x^2 + 4x + 4 = 9
(x + 2)^2 = 9
sqrt( (x+2)^2 ) = sqrt(9)
(x + 2) = +- 3
(x +2) = 3 or (x + 2) = -3
x = 1 or x=-5 Let's try one together. Solve x^2 -8x + 2 = 5 b = _____? b/2 = _____ (b/2)^2 = _____ b = -8 b = -4 b = 16 x^2 -8x + ____ + 2 = 5 +_______ x^2 -8x + ____ + 2 = 5 +_______
x^2 -8x + ____ = 5 -2 +_______
x^2 -8x + 16 = 5 -2 + 16
(x __ ____)^2 = 19 Fill in the blanks. Raise your hand if you did not get
x = 4 - sqrt(19)
x = 4 + sqrt(19) Let's try a harder problem.... Solve 6x^2 -7x -3 = 0 1. Get rid of the 3
6x^2 - 7x = 3 2. Get the expression in the form x^2 + bx
x^(2) - (7/6)x = 3/6 3. Complete the square
b = (-7/6)
b/2 = (-7/6)*(1/2)
(b/2)^2 = (-7/12)(-7/12)
(b/2)^2 = 49/144 4. Add (b/2)^2 to both sides of the expression
x^2 -7/6x + 49/144 = 1/2 + 49/144 5.Factor left side
(x - 7/12)^2 = 72/144 + 49/144
(x - 7/12)^2 = 121/144 7.Solve
sqrt( (x - 7/12)^2) = sqrt( 121/144)
sqrt( (x - 7/12)^2) = sqrt(121)/sqrt(144)
x - 7/12 = +- 11/12
x= 3/2 or x = -1/3 Try one on your own now. (5 minutes) Complete Squares twice.
9x^2 + 4y^2 + 54x - 8y + 49 HINT
9x^2 + 54x +______+4y^2 - 8y +______+ 49=0 + ______ + _______ CLOSURE Being able to complete the square will help you solve conics problems in future lessons.
The last problem you saw, resembles the types of problems you will be seeing
Keep practicing completing squares and solving problems. Classwork!!! Page 257 #s 11,15,17,23
Page 310 #s 31,33 As you guys know, the reason I teach lessons like this
is because I'm a teacher in training, as such, I'm required to
teach several math lessons to you guys.
In order for me to improve, I need to get feedback from all of you.
So please, using half a sheet of paper, PLEASE respond to the following
questions as honest as you can be. You don't need to write your name.
THANK YOU! STUDENT FEEDBACK 1. From a scale from 1 to 10, how would you rate this lesson?
2. What did you like the MOST about today's lesson?
3. What did you like the LEAST about today's lesson?
4. Did you already know how to complete the square prior to this lesson?
5. After today's lesson, do you feel you can complete the square of any expression?
6. Did you like the way today's lesson was presented?
7. Would you like to see future lessons presented using the same format as today's lesson?
8. Any other comments you might have are always welcome AND NOW...IT'S TIME....FOR SOME............ Standards 8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.
17.0 Given a quadratic equation of the form ax^2 +by^2 +cx +dy +e =0,
students can use the method for completing the square to put the
equation into standard form and can recognize whether the graph of the
equation is a circle, ellipse, parabola, or hyperbola. Students can then
graph the equation. IF YOU GUESSED...... Preview of Questions to Come! Some of these equations, will turn into the following pictures.... Does that mean GRAPHING? Aaaaaahhh!!!!!!! How much graphing.......exactly???