6.1a Completing the Square

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6.1a Completing the Square
Objective To create perfect square trinomials.
Perfect Square Trinomials These are the
solutions to (a b)2 and (a b)2.
1. (x 1)2
Note What pattern do you see with b and c (in
ax2 bx c) with respect to the constants from
each binomial?
2. (x 2)2
3. (x 3)2
4. (x 4)2
5. (x 5)2
(a b)2 and (a b)2
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What number makes each a perfect square
trinomial? How did you determine this number?
Factor each.
x2 10x _____
x2 14x _____
x2 - 20x _____
x2 - 50x _____
Now try these.. x2 x _____ x2 7x
_____ x2 11x _____
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Complete the square if a 1
y x2 2x 7

• Move the constant to the other side.

• What number completes the square?

y - 7 x2 2x ____

• Add this number to both sides.

• Factor the trinomial.

y 7 x2 2x

• Move the constant back.

y (x )2
This is also called vertex form.
y
You try these y x2 4x 9 y x2 5x
1 y x2 9x 3
Summarize this procedure.
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y 2x2 4x - 3
Move the constant to the other side.
Factor the a from the x2 and x term.
Complete the square.
Add a . (this value) to the other side and
factor the trinomial.
Move the constant back.
You try these y 3x2 12x 4 y 2x2
3x 2 y 3x2 x 4 y
-2x2 8x 3 y -6x2 5x 1
Assign WS
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WARM-UP

• 1. Solve x2 6x - 16 0

2. What are these 2 things that we just found??
3. Put the equation in 1 into vertex form.
4. What is the vertex? Now sketch!
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Worksheet Answers
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6.1b Solve a Quadratic by Completing the
Square
Objective To solve a quadratic by completing
the square
To solve a quadratic means to find the
x-intercepts, the roots, the zeros. What does
this mean?
What methods have we used to solve quadratics?
Factoring x2 6x 16 0
Quadratic formula 2x2 x 2 0
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We can also solve the quadratic from vertex form.
(Completing the Square)
2(x 1)2 4 0

• Solve for ( )2 .

• Take the square root of both sides. Dont
  forget the !!

• Solve for x.

• This is the answer you would get if you had used
  the quadratic formula. (This one is not
  factorable.)

Complete the square to solve these (2x 5)2
5 0 x2 4x 9 0 3x2 9x 12 0
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Solve over the set of complex numbers
What does this mean?
x2 x 4 0 x2 9x 3 0 x2 3x 8
0
Assign 6.1b 1-55 odd
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6.2a Quadratic Formula
Objective To solve quadratic equations in
various forms

• Solve ax2 bx c 0 by completing the
  square.

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Now, lets solve some equations!

• Make sure
• quadratic 0
• get rid of fractions by multiplying by LCD
• try factoring first
• apply quadratic formula / complete the square

Ex. 2 .02x2 - .03x .05 0
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Ex. 3 (x 3)(x 1 ) 5
Note All complex roots come in pairs. If you
have x 3 i as one root, you will also have x
________ as the other root.
Ex 5. x3 64 0
Assign 6.2a 1-35 odd
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6.2b Finish Completing the Square/Applications
If you are given x 2i as on complex root, you
know x _____ is another root. If x 1 3i is
one root, then x is another root.
Write an equation for a polynomial with roots x
2 and 3i
1st You know for each complex root, there is
another. 2nd Put the roots back into factored
form. 3rd Expand.
y
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Solve y x4 x3 3x2

How many roots could this have?
solve
---Applications---
Pg 514 52 ? s(t) 16t2 where s(t)
distance and t time
How do you know where to substitute the 100?
Should you keep both answers?
Find the height after 1 sec?
Is this the height from the ground or from the
top of the cliff?
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Pg 515 54 h(t) 16 32t 16t2
1. When will the coin reach its original height?
2. What is the maximum height of the coin?
3. Put the equation into vertex form? What do
you notice about your answer and the answer to
question 2?
4. When will the coin hit the ground?
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2150 (300x - .5x2) (200x 1600)
Must 0 Distribute the negative.
One More!! How would you solve ix2 3x
4i 0 ?
Assign 6.2 37-49 odd, 51-59 odd, 63-65, 67-73
odd, 75, 77
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6.2b Solutions

• 64.

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6.3 The Discriminant
Objective To understand the use of the
Discriminant and to write equations given
the roots of the equations
The discriminant tells the number and type of
solutions to a quadratic, but not the solutions.
It can also tell you if the quadratic is
factorable.
D b2 - 4ac
Where is this formula from?
Ex1 y x2 6x 9 Ex2 y x2 x
3 Ex3 y x2 6x 16 Ex4 y x2 2x 5
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Summary
If D perfect square, then
How many solutions can a perfect square trinomial
have? x2 6x 9 0 4x2 12x 9 0
(x 5)2 0
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Find k so that the quadratic has one solution.
(What does this mean?)
Ex1 x2 kx 36 0 Ex2 x2 kx 12
0 Ex3 2x2 12x k 0 Ex4 kx2 6x 4 0
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Write the equation for the polynomial with the
given roots.
Ex7 x 3, -4i
Ex8 x 2 i, 5
Ex x 5 is a solution of multiplicity 1 and x
-1 is a solution of multiplicity 2. Write an
equation for the polynomial.
Assign 6-3 1-48 (x3), 49-55 odd, 71, 73, 83
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WARM-UP

• Find the missing values of b and c.
• (x 3)2 x2 bx c
• (x c)2 x2 bx 25
• (x b)2 x2 8x c
• Which has a possibility of more than one solution?

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6.3 Solutions

• 6. D -8 2 complex solutions
• 12. D 89 2 irrational solutions
• 18. K 25
• 24. x2 7x 10 0
• 30. 3x2 16x 5 0
• 36. 21a2 5a 4 0
• 42. x2 3 0
• 48. x2 2x 2 0

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6.4 Other Equations
Objective To use our knowledge of solving
quadratics to solve other equations.
You solved 2x2 7x 3 0 in the warm up. How
could you use this knowledge to solve the
following problems?
Ex1 2(x - 3)2 7(x - 3) 3 0
Ex3 x5 3x3 4x 0
Could you use the QF to solve Ex3? How?
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Applications
g, vo, and ho are constants and will be
substituted with given values. g gravity
meters use 9.8m/s2 feet use 32ft/s2 vo
starting velocity and ho starting height

• Ex4 A ball is dropped out of a 40 ft window.
• g , vo , ho
• Write an equation for h(t).
• When is the ball half way to the ground?
• When does the ball hit the ground?
• What values of time are valid for this situation?

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• Ex5 Fred throws Pebbles into the air at 20
  ft/s. If he releases her at a height of 5 feet,
• Find g _____, vo _____ and ho _____
• b. Write an equation for h(t).
• c. How high will Pebbles go?
• d. When will she be back to a height of 5?
• e. If Fred misses catching her at 5 feet, when
  will she hit the
• ground? Oops!!!

41 The arch is not a parabola it is
classified as a catenary curve since it is made
of pieces. A 693.8597 68.7672cosh.0100333x
is the actual equation for the St. Louis Arch.
Assign1-35 odd, 42, 43, 44, 57-63 odd OH I
like questions 43 and 44.?
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6.4 Solutions

• 42. x 14/11 44. l 2w 162
• Area 25.8
• 44. a. l 2w 162
• b. A -2w2 162w
• c.
• d. Maximum area 3,280.5 yd2

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6.5 Graphing Parabolas

• Objective At the end of the hour you should be
  able to graph a parabola using the roots,
  vertex, axis of symmetry and a value.

Lets see what we already know that will help us
graph the following parabolas.
Ex. 1 Given y (x - 5)(x 1)
Roots
Axis of symmetry
Vertex
What direction will it open?
a
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Ex. 2 y -x2 2x 5
Roots Axis of symmetry Vertex Direction a
So far, in order to find the vertex what have we
done?
Guess what, Ive been keeping a really BIG
secret!!!
GASP!!
There is another way to find the vertex.
The x-value of the vertex is How do you
find the y-value?
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Roots Axis of symmetry Vertex Direction a
Vertex form
So.how does the value of a affect the graph??
Up? Down?
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Lets examine some quadratics using the graphing
calculator.
Compared to y x2, when will the
quadratic Become narrow? Widen?
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Now how does this help us graph the parabola??
y (x 3)2 - 2
Vertex? Locate and label.
Value of a?
From the vertex, you only need to graph y ax2.
Now try y 2(x 1)2 5 Vertex? Value
of a? Graph.
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Graph these using vertex form.
Ex1 y (x 1)2 3

• Vertex?
• a
• Opens?
• Roots?
• Graph
• Axis of symmetry?

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Graph these using vertex form.
Ex2 y 2(x 1)2 - 4

• Vertex?
• a
• Opens?
• Roots?
• Graph
• Axis of symmetry?

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Graph these using vertex form.
Ex3 y - ½ (x - 3)2 4

• Vertex?
• a
• Opens?
• Roots?
• Graph
• Axis of symmetry?

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Graph these using vertex form.
Ex4 y -3(x - 2)2 8

• Vertex?
• a
• Opens?
• Roots?
• Graph
• Axis of symmetry?

Assign 6.5a Make sure you give the vertex,
roots, value of a, direction opens, axis of
symmetry and graph.
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6.5a Solutions
14. V(-1, -3)

• V(3, 1)
• x 2 and x 4

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20. V(1, 4) x 0 and x 2
18. V(3, -3) x 0 and x 6
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6.6 Quadratic Inequalities
Objective To understand what it means for f(x)
gt 0, f(x) lt 0 , f(x) 0 and to solve
and graph quadratic inequalities.
f(x) gt 0 means y gt 0
f(x) lt 0 means y lt 0
f(x) 0 means y 0
So.. x2 x - 6 gt 0
To solve this we need to find the roots.
lt and gt use an open circle lt and gt use
a closed circle
How do you determine where to shade?
Test points? Know the Graph?
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Now try these.
Ex1 x(x 1)2(x 3)(x 4) gt 0
Assign 6.6 1-37 odd, 41, 43, 49, 51, 57, 59