Sargent Fall 2010 Algebra 1 Notes: Chapter 9
Name:
Date:__________ Period: __________
CHAPTER 9: Quadratic Equations and Functions
Notes #23
9-1: Exploring Quadratic Graphs
A. Graphing 2y ax
A ____________________ is a function that can be written in the form 2y ax bx c where
a, b, and c are real numbers and a 0.
Examples: 25y x 2 7y x 2 3y x x
The graph of a quadratic function is a U-shaped curve called a ________________. When
graphed it will look like: OR
You can fold a parabola so that the two sides match exactly. This property is called:
_____________.
The highest or lowest point of the parabola is called the ________________, which is on the
axis of symmetry.
B. Identifying a Vertex
Identify the vertex of each graph. Tell whether it is a minimum or a maximum.
1.) 2.)
Vertex: ( , ) _________ Vertex: ( , ) _________
3.) 4.)
Vertex: ( , ) _________ Vertex: ( , ) _________
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Graph each function. State the domain, the vertex (min/max point), the range, the x-intercepts,
and the axis of symmetry.
5.) f(x)= x2 – 4
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
6.) h(x) = -2x2
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
7.) f(x) = 21
2x
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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8.) k(x) = x2 + 2x + 1
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
9.) f(x) = x2 – x – 6
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
C. Comparing Widths of parabolas
The value of a, the coefficient of the 2x term in a quadratic function, affects the width of the parabola
as well as the direction in which it opens.
When 1,a then the parabola is steeper, (or _________) than y = x2
When 1,a then the parabola is not as steep, (or _________) than y = x2
Order each group of quadratic functions from widest to narrowest graph:
10.) 2 2 21( ) 3 , ( ) 4 , ( )
2f x x f x x f x x 11.) 2 2 25
2 , , 4
y x y x y x
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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D. Applications
12.) A monkey drops an banana from a branch 64 feet above the ground. Gravity causes the banana to
fall. The function 216 64h t gives the height of the banana, h, in feet, after t seconds.
a) Graph this quadratic function b) When does the banana hit the ground?
feet
time (sec)
70
60
50
40
30
20
10
321
13) A bungee jumper dives from a platform. The function h = -16t2 + 160 describes her height, h,
after t seconds in the air.
a) What will her height be after 1 second? b) what will her height be after 2 seconds?
c) How far did she fall between 1 and 2 seconds in the air?
t 2( ) 16 64h t t (t, h(t))
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Notes #24
9-2: Quadratic Functions
y = ax2 + bx + c
One key characteristic of a parabola is its vertex (min/max point). Yesterday we found the vertex after
we graphed the function. It would help to find the vertex first.
Vertex
- find x = 2
b
a
- plug this x-value into the function (table)
- this point (___, ___) is the vertex of the parabola
Graphing
- put the vertex you found in the center of
your x-y chart.
- choose 2 x-values less than and 2 x-values more
than your vertex.
- plug in these x values to get 4 more points.
- graph all 5 points
Find the vertex of each parabola. Graph the function and find the requested information
1.) f(x)= -x2 + 2x + 3 a = ____, b = ____, c = ____
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Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
2.) h(x) = 2x2 + 4x + 1
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Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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3.) k(x) = 2 – x –1
2x
2
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Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
Without graphing the quadratic functions, complete the requested information:
4.) 2( ) 3 7 1f x x x
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
5.) 25( ) 3
4g x x x
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
6.) 2211
3y x
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
7.) 20.6 4.3 9.1y x x
What is the direction of opening? _______
Is the vertex a max or min? _______
Wider or narrower than y = x2 ? ___________
B. Application
8.) Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an
initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How
far above the ground will it be? The equation 216 88 520h t t gives the star’s height h in feet at
time t in seconds.
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Notes #25
9-3: Finding and Estimating Square Roots
A. Finding Square Roots
The expression __________ means the positive, or __________ square root.
The expression __________ means the negative square root.
The expression ___________ means both the ____________ and _____________ square root
Simplify each expression.
1.) 64 2.) 100 3.) 9
16
4.) 0 5.) 25 6.) 0.09
7) 27 8.) 72 9.) 108
10.) 4
5 11.)
45
4 12.) 2.25
B. Rational and Irrational Square Roots
Tell whether each expression is rational or irrational.
13.) 144 14.) 1
5 15.)
1
9 16.) 7
17.) Between what two consecutive integers is 28.34 ?
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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18.) Between what two consecutive integers is 68.7 ?
19.) Between what two consecutive integers is 14.3 ?
C. Application: Pythagorean Theorem (Review)
Use the Pythagorean theorem (_________________) to solve for the missing side of the right
triangle.
20.) 21.)
x8
4
y
46
9-4: Solving Quadratic Equations
A. Solving Quadratic Equations by Graphing
The solutions of a quadratic equation and the related x-intercepts are often called _______ of
the equation or _______ of the function.
1.) The function f(x) = x2 + x – 6 is graphed to the left.
a) Circle and name the zeros of the function graphed here.
( , ) and ( , )
b) Use this graph to solve the equation: x2 + x – 6 = 0
(This is asking: “At what x-values does y = 0?”)
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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B. Solving by Graphing
Solve each equation by graphing the related function:
Find the vertex and 4 other points on the parabola; graph.
Find the x-intercepts from the graph. These are the _______ or _______.
2.) x2 – 4 = 0
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3.) 2x2 – 2 = 0
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4.) x2 + 6 = 0
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Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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C. Solving Quadratic Equations Using Square Roots
Isolate the variable or expression being squared (get it ______________)
Square root both sides of the equation (include + and – on the right side!)
This means you have _____________ equations to solve!!
Solve for the variable (make sure there are no roots in the denominator)
5.) x2 = 25 6.) 3x
2 = 48
7.) 4x2 – 1 = 0 8.) 3m
2 – 5 = 0
9.) 2y2 – 81 = 0 10.) 36b
2 – 7 = 0
11.) (x – 1)2 = 4 12.) (2y + 3)
2 = 49
13.) (r + 5)2 = 12 14.) (3m – 1)
2 = 20
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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If the left side is not already factored or squared, _______________ it!
15.) x2 + 2x + 1 = 8 16.) n
2 – 14n + 49 = 3
17.) w2 + 22w + 121 = 169 18.) g
2 + 10g + 25 = 18
D. Application
19.) A museum is planning an exhibit that will contain a large globe. The surface area of the globe will
be 100 ft2. Find the radius of the sphere producing this surface area. Use the equation 24S r ,
where S is the surface area and r is the radius.
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Notes #26
9-5: Solving Quadratic Equations by Factoring
A. Solving Quadratic Equations
Zero Product Property
List some pairs of numbers that multiply to zero:
(___)(___) = 0 (___)(___) = 0 (___)(___) = 0 (___)(___) = 0
What did you notice? _______________________________________________
Use this pattern to solve for the variable:
1. get = 0 and factor (sometimes this is done for you)
2. set each ( ) = 0 (this means to write two new equations)
3. solve for the variable (you sometimes get more than 1 solution)
1.) (3)(x) = 0 2.) (2)(x + 1) = 0 3.) -2y(y – 7) = 0
4.) (m + 1)(5m – 3) = 0 5.) 3
2 9 05
w w 6.) 2 6 4 8
03 7 5 9
x x
7.) x2 – 4x – 5 = 0 8.) y
2 + 6 + 5y = 0 9.) 4v
2 – 9 = 0
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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10.) 2 42 0x x 11.) 23 2 21x x 12.) 22 15x x
13.) v(v + 3) = 10 14.) b(b – 2) = 3(b + 2)
B. Solving Word Problems with Quadratics
Steps:
1. Draw a picture and define your variable (let statement)
2. Write an equation
3. Get = 0 (bring all variables and numbers to one side)
4. Factor completely and solve
5. Do all the answers make sense?
6. Write your answer in a complete sentence
Translate and solve:
15.) The square of a positive number minus twice the number is 48. Find the number.
Let n = _____________ _________ - _________ = ______
16.) One more than a negative number times one less than that number is 8. Find the number.
Let n = ______________ (_________)(_________) = _____
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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17.) The product of two consecutive integers is 12. Find the integers.
Let x = 1st integer
______ = 2nd
integer
18.) The product of two consecutive odd integers is 35. Find the integers.
Let x = 1st odd integer
______ = 2nd
odd integer
19.) The length of a rectangle is 3ft greater than its width. The area of the rectangle is 54ft2. Find the
length and the width of the rectangle.
20.) The area of a square is 5 more square inches than there are inches in the square’s perimeter. Find
the length of a side of the square.
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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21.) Two less than the square of a number is equal to the number. Find the number.
___________ - _________ = _______
22.) The sum of the square of a number and three times the number is the same as one less than the
number. Find the number.
__________ + _________ = _________ - _________
Notes #27
9-6: Completing the Square
So far in this course, we have solved quadratics by _______________, __________________ and
___________________. We will eventually learn two more ways to solve quadratics.
Solve these equations. What makes these quadratics “easy” to solve?
a) (x – 1)2 = 9 b) (k + 2)
2 = 12
Solving quadratics by _________________ ______ _______________ helps us turn all quadratics
into this form.
Complete the square:
Take half the b (the x coefficient)
Square this number (no decimals – leave as a fraction!)
Add this number to the expression
Factor – it should be a binomial, squared ( )2
1.) x2 + 6x + _____ 2.) m
2 – 14m + _______
( )( )
( )2
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Find the value of n such that each expression is a perfect square trinomial:
3.) w2 + 7w + n 4.) k
2 – 5k + n
5.) j2 – j + n 6.) y
2 + 18y + n
Solving by Completing the Square:
Collect variables on the left, numbers on the right
Divide ALL terms by a; leave as fractions (no decimals!!)
Complete the square on the left – add this number to BOTH sides
Square root both sides (include a ______ and _______ equation!)
Solve for the variable (simplify all roots)
7.) x2 + 4x – 5 = 0 8.) x
2 – 6x – 11 = 0
9.) k2 – 4k – 7 = 0 10.) m
2 – 5m + 1 = 0
11.) 2y2 + 6y – 18 = 0 12.) 2x
2 – 3x – 1 = 0
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Cumulative Review: Solving Quadratics
Solve by factoring:
1.) 12k2 – 5k = 2 2.) 49m
2 – 16 = 0
Solve by using square roots:
3.) 4w2 = 18 4.) 3y
2 – 8 = 0
5.) 5m2 – 16 = 0 6.) (2x – 1)
2 = 20
Solve by completing the square:
7.) x2 – 10x + 7 = 0 8.) 4m
2 + 12m – 7 = 0
9.) 3y2 – 2y – 1 = 0 10.) 2x
2 – 20x = -50
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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11.) 5x2 + 13x + 7 = 0 12.) ax
2 + bx + c = 0
Notes #28
9-7: Using the Quadratic Formula
A. Review of Simplifying Radicals and Fractions
Simplify expression under the radical sign, reduce
Reduce only from ALL terms of the fraction
1.) 6 18
2
2.)
5 20
2
3.) 4 20
4
4.) 8 27
2
5.) 29 ( 5) (4)(2)(3)
4
6.)
29 (6) 4( 3)( 3)
4
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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B. Solving Quadratics using the Quadratic Formula
So far, we have solved quadratics by: (1) _______________, (2) ______________,
(3) ___________________, and (4) _________________
The final method for solving quadratics is to use the quadratic formula.
Solving using the quadratic formula:
Put into standard form (ax2 + bx + c = 0)
List a = , b = , c =
Plug a, b, and c into
2 4
2
b b acx
a
Simplify all roots, reduce
Solve by using the quadratic formula:
1.) x2 + x = 12
2 4
2
b b acx
a
(std. form):
a = _____
b = _____
c = _____
2.) 5x2 – 8x = -3
2 4
2
b b acx
a
(std. form):
a = _____
b = _____
c = _____
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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3.) 2x2 = 4 – 7x 4.) 3x
2 – 8 = 10x
5.) -x2 + x = -1 6.) 3x
2 = 7 – 2x
Review of Solving Quadratics:
Solve by factoring:
7.) 4m2 +5m – 6 = 0 8.) 3x
3 – 27x = 0
Solve by using square roots:
9.) 4b2 – 1 = 0 10.) 3y
2 – 36 = 0 11.) (3x + 1)
2 = 18
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Solve by completing the square:
12.) x2 – 10x + 9 = 0 12.) x
2 – 7x – 18 = 0
13.) 4m2 + 12m + 5 = 0 14.) 3y
2 + 2y – 1 = 0
Solve by using the quadratic formula:
15.) x2 – 20 = 0 16.) x
2 – 6x + 9 = 0
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Notes #29
9-8: Using the Discriminant
Quadratic equations can have two, one, or no solutions. You can determine how many solutions
a quadratic equation has before you solve it by using the ________________.
The discriminant is the expression under the radical in the quadratic formula:
2 4
2
b b acx
a
Discriminant = b2 – 4ac
If b2 – 4ac = 0, then the equation has 1 solution
If b2 – 4ac < 0, then the equation has 0 real solutions
If b2 – 4ac > 0, then the equation has 2 solutions
A. Finding the number of x-intercepts
Determine whether the graphs intersect the x-axis in zero, one, or two points.
1.) 24 12 9y x x 2.) 23 13 10y x x
B. Finding the number of solutions
Find the number of solutions for the following:
3.) 23 5 1x x 4.) 2 3 7x x
5.) 9x2 – 6x = 1 6.) 4x
2 = 5x + 3
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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C. Review of Solving Quadratics
Solve by factoring:
7.) 2x2 + 12x = -10 8.) 16(x – 1) = x(x + 8)
Solve by using square roots:
9.) 3b2 – 1 = 7 10.) (3x + 1)
2 = 18
Solve by completing the square:
11.) x2 – 10x – 11 = 0 12.) x
2 – 3x – 6 = 0
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Solve using the quadratic formula:
13.) 6x2 + 7x = 5 14.) x
2 = 8 – 6x
Graph the quadratic. Name the vertex, axis of symmetry, x-intercepts, domain, and range.
13.) f(x)= x2 – 9
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Notes #30: Solving Radical Equations with Quadratics (Section 10.4)
Solving Radical Equations:
Isolate the _________________
______________ both sides. If one side is a binomial, be sure to use ___________ to
square it.
Get all terms to one side to = 0
Solve the quadratic using: factoring, quadratic formula, or completing the square.
Check your solution by ___________________ into the original equation. Check for
extraneous roots.
1.) 2 3 1x
2.) 4 2 5x
3.) 2x x 4.) 35 2x x
5.) 6 9x x 6.) 11 28x x
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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7.) 2 7 2x x 8.) 3 2 4x x
9.) Graph the quadratic. Find the requested
information: f(x)= -x
2 – 3x + 4
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Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Chapter 9 Review (Classwork #31) Solve by factoring: 1.) 4x2 – 12x = 0 2.) 6x2 + 15x = 21 Solve by square rooting: Solve and check:
3.) 5y2 – 6 = 12 4.) 1 1x x
Solve by completing the square: Solve using the quadratic formula: 5.) x2 – 6x – 27 = 0 6.) x2 – x – 1 = 0 Find the discriminant; find the Define a variable, write an equation, number of real solutions: and solve: 7.) 4x2 – 6x + 3 = 0 8.) The width of a rectangle is two inches less
than its length. The area of the rectangle is 63in2. Find the dimensions of the rectangle.
Find the vertex of each parabola. Graph the function and find the requested information 9.) f(x)= x
2 – 2x – 3 a = ____, b = ____, c = ____
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x
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Vertex: _______
Max or min? _______
Direction of opening? _______
Wider or narrower than y = x2 ?
_______________
Domain: ________
Range: ________
x-intercepts: ____________
Axis of symmetry: ________
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Algebra 1: Chapter 9 Study Guide Name: ___________________________
Match each graph with its function
1.) 2.) 3.)
Without graphing, provide the requested information about each parabola:
4.) a = ______ b=______ c=________
vertex _________
equation of the axis of symmetry ________
max or min? _______
direction of opening? _______
wider or narrower than ______
5.) a = ______ b=______ c=________
vertex _________
equation of the axis of symmetry ________
max or min? _______
direction of opening? _______
wider or narrower than ______
Graph the function. State the domain, the vertex (min/max point), the range, the
x-intercepts, and the axis of symmetry.
6.) y = x2 – 4x + 3
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
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Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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7.) y = -x2 – 6x – 5
Domain: ________
Range: ________
Vertex:________
Max or min?________
x-intercepts: ____________
Axis of symmetry: ________
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y
Solve by factoring
8.) k2 – 3k – 10 = 0 9.) 3m
2 – 2m = 21
Solve by using square roots
10.) 3m2 – 108 = 0
11.) 4w2 – 75 = 0
Solve by completing the square
12.) x2 + 4x – 5 = 0 13.) x
2 – 6x – 3 = 15
14.) 2x2 + 6x – 18 = 0
Solve by using the quadratic formula: Write down the formula:
15.) 2x2 + 5x + 3 = 0 16.) x
2 – 2x = 2
Sargent Fall 2010 Algebra 1 Notes: Chapter 9
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Write the expression for the discriminant. Use this to find the number of real solutions for each
equation: 17.) 3x
2 + 6x + 7 = 0 18.) x
2 + 5x = -2 19.) x
2 – 4x + 4 = 0
Solve each equation. Check for extraneous solutions.
20.) 1 1x x 21.) 3 2x x
Translate and solve:
22.) One more than a positive number times one
less than that number is 8. Find the number.
23.) The product of two consecutive odd integers
is 63. Find the numbers.
24.) The length of a rectangle is one inch longer than twice the width. The area of the rectangle is
36in2. Find the length and width of the rectangle.