Download Test 2 Review - Solving and graphing Quadratics

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Review Test 2: Quadratic Solving and Graphing
Solving quadratics by Factoring
A. Factoring Quadratics
Examples of monomials:_______________________________
Examples of binomials:________________________________
Examples of trinomials:________________________________
Strategies to use: (1) Look for a GCF to factor out of all terms
(2) Look for special factoring patterns as listed below
(3) Use the X-Box method or the Grouping method
(4) Check your factoring by using multiplication/FOIL
Factor each expression completely. Check using multiplication.
1.) 3 x 2  15 x
2.) 6 x 2  24
4.) 25 x 2  81
5.) m2  22m  121
3.) x 2  5 x  24
6.) 4 x 2  12 x  9
1
7.) 5 x 2  17 x  6
8.) 3x 2  5 x  12
9.) 25t2  110t + 121
10.) 16 x 2  36
11.) 9a 2  42a  49
12.) 6 x 2  33x  36
B. Solving quadratics using factoring
To solve a quadratic equation is to find the x values for which the function is equal to _____. The solutions are
called the _____ or _______of the equation. To do this, we use the Zero Product Property:
Zero Product Property
List some pairs of numbers that multiply to zero:
(___)(___) = 0
(___)(___) = 0
(___)(___) = 0
(___)(___) = 0
What did you notice? _______________________________________________
ZERO PRODUCT PROPERTY
If the _________ of two expressions is zero, then _______ or _______ of
the expressions equals zero.
Algebra
If A and B are expressions and AB = ____ , then A = _____ or B = __.
Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is,
x = __________ or x = _________.
Use this pattern to solve for the variable:
1. get the quadratic = 0 and factor completely
2. set each ( ) = 0 (this means to write two new equations)
3. solve for the variable (you sometimes get more than 1 solution)
Find the roots of each equation:
13.) x 2  7 x  30  0
6  4
8
2
14.)  x   x    0
7  5
9
3
Find the zeros of each equation:
17.) v(v + 3) = 10
16.) 2 x 2  8 x  30  x  34
15.) 3x 2  2 x  21
18.) 2 x 2  x  15
Find the zeros of the function by rewriting the function in intercept form:
19.) y  6 x 2  3x  63
20.) f  x   12x2  6x  6
21.) g  x   49x2 16
Converting between forms:
From intercept form to standard form

Use FOIL to multiply the binomials
together

Distribute the coefficient to all 3 terms
From vertex form to standard form

Re-write the squared term as the
product of two binomials

Use FOIL to multiply the binomials
together

Distribute the coefficient to all 3 terms

Add constant at the end
Ex:
y  2  x  5 x  8
Ex: f  x   4  x  1  9
2
Graph the function. Label the vertex and axis of symmetry:
22.) y  2  x  1 ( x  3)
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
Vertex: _______
Maximum or minimum value: _______
x-intercepts: _______
Axis of symmetry: ________
Compare width to the graph of y = x2
_________________________
-8
-9
-10
23.) k(x) = 2 – x –
1 2
x
2
Vertex: _______
y
10
Max or min? _______
9
8
7
6
Direction of opening? _______
5
4
3
Axis of symmetry: ________
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Compare to the graph of y = x2
-2
-3
-4
-5
_________________________
-6
-7
-8
-9
-10
24.) h(x) = 2x2 + 4x + 1
y
Vertex: _______
10
9
8
Max or min? _______
7
6
5
4
Direction of opening? _______
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Axis of symmetry: ________
-2
-3
-4
-5
Compare to the graph of y = x2
-6
-7
-8
-9
-10
_________________________
Solve Quadratic Equations by Finding Square Roots
C. Simplifying Square Roots:
 Make a factor tree; circle pairs of “buddies.”
 One of each pair comes out of the root, the non-paired numbers stay in the root.
 Multiply the terms on the outside together; multiply the terms on the inside together
Simplify:
1.) 3 120
2.) 5 72
D. Multiplying Square Roots:
 Simplify each radical completely by taking out “buddies”
 (outside • outside) inside inside or (a b )(c d )  ________
 Simplify your answer, if possible
Simplify:


3.) 5 12 3 15



4.) 2 5 2 5

E. Simplifying Square Roots in Fractions:
a
2
2
2
 Split up the fraction:



b
25
5
25
 Simplify first by taking out “buddies” or reducing (you can only reduce two numbers that are both
under a root or two numbers that are both not in a root)
 Square root top, square root bottom
 If one square root is left in the denominator, multiply the top and the bottom by the square root and
simplify
 Reduce if possible
Simplify:
6.)
7.)
7
5.)
3
F. 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)
8.) x2 = 25
10.)
9.) 3x2 = 81
4x2 – 1 = 0
11.)
m2
3  2
15
13.) 3  x  2   6  34
2
12.) (2y + 3)2 = 49
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 =
b  b2  4ac
x
2a

Plug a, b, and c into

Simplify all roots (look for
1  i ); reduce
G. Solve by using the quadratic formula:
2
1.)
x + x = 12
x
b  b2  4ac
2a
(std. form):
a = _____
b = _____
c = _____
2.) 5x – 8x = -3
2
b  b2  4ac
x
2a
(std. form):
a = _____
b = _____
c = _____
3.) -x2 + x = -1
4.) 3x2 = 7 – 2x
5.) -x2 + 4x = 5
6.) 4( x  1)2  6 x  2
Using the Discriminant
 Quadratic equations can have two, one, or no solutions (x-intercepts). 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: x 
b  b2  4ac
2a
Discriminant = b2 – 4ac
If b2 – 4ac < 0, then the equation has 2 imaginary solutions
If b2 – 4ac = 0, then the equation has 1 real solution
If b2 – 4ac > 0, then the equation has 2 real solutions
H. Finding the number of x-intercepts
Determine whether the graphs intersect the x-axis in zero, one, or two points.
1.) y  4 x 2  12 x  9
2.) y  3x 2  13x  10
I. Finding the number and type of solutions
Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.
3.) 3 x 2  5 x  1
4.) x 2  3 x  7
5.) 9x2 – 6x = 1
6.) 4x2 = 5x + 3
J. Graph each function by making a table of values with at least 5 points. (A) State the vertex. (B)
State the direction of opening (up/down). (C) State whether the graph is wider, narrower, or the
same width as y  x 2 .
1.)
f ( x) 
1
( x  6) 2  5
2
y
10
9
8
7
6
5
4
Vertex: _______
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Direction of opening? _______
-2
-3
-4
Compare width to the graph of y = x2
_________________________
-5
-6
-7
-8
-9
-10
2.) k(x) = x2 + 2x + 1
y
10
9
8
7
6
5
4
3
Vertex: _______
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
Direction of opening? _______
-2
-3
-4
Compare width to the graph of y = x2
_________________________
-5
-6
-7
-8
-9
-10
K. Evaluate the following expressions given the functions below:
g(x) = -5x - 3
a. g(8) =
d. j(
2
)=
3
h. Find x if g(x) = 41
f(x) = 2x2 + 9
b. f(-4) =
h( x ) 
8
8 x
j ( x)  2 x  7
c. h(–5) =
1
)
2
e.
g(4a +
i.
Find x if h(x) = –2
f.
f(h(-16))
j. Find x if g(j(x)) = -29
Methods for Solving Quadratic Equations: SUMMARY
A.) Factoring
1st: Set equal to 0
2nd: Factor out the GCF
3rd: Complete the X & box method to find the factors
4th: Set every factor that contains an x in it, equal to 0 & solve for x.
B.) Completing the Square
1st: Move the constant (number with no variable) to the right so that all variables are on the left
& all constants are on the right.
nd
2 : Divide every term in the equation by the value of a, if it is not already 1.
2
b
3rd: Create a perfect square trinomial on the left side by adding   to both sides.
2
2
b

4th: Factor the left side into a  x   and simplify the value on the right side.
2

th
5 : Take the square root of both sides of the equation. REMINDER: Don’t forget the 
6th: Solve for x
C.) Finding Square Roots
1st: Isolate the term with the square.
2nd: Take the square root of both sides of the equation. REMINDER: Don’t forget the 
3rd: Solve for x.
D.) Quadratic Formula
1st: Set the equation equal to 0.
2nd: Find the values of a, b, and c & plug them into the Quadratic Formula:
b  b2  4ac
2a
3rd: Simplify the radical as much as possible.
4th: If possible, simplify the numerator into integers.
x
46 3
), divide BOTH
2
terms by the number in the denominator (the example would result in 2  3 3 )
5th: Divide. REMINDER: If you have 2 terms in the numerator (ex:
WORKED OUT QUADRATIC PROBLEMS
Real-World Application
Use the following problems to have students determine the best method to use to solve them.
A sprinkler sprays water that covers a circular region of 75 square feet. Find the diameter of the circle.
A  r
2
75  r
75  r
2
 75  r
5 3r
2
Since a diameter cannot be negative, use the positive
square root. The diameter is twice the radius.
The diameter of the circle is 10 3  17 ft .
The length of a rectangle is three more than twice the width. Determine the dimensions that will give
a total area of 27 m2.
27  2 w  3w 
2
27  2 w  3w
2
0  2 w  3w  27
0  2 w  9 w  3
0  2 w  9 and 0  w  3
w
9
and w  3
2
Since w represents the width of a rectangle we must omit the negative value. Therefore,
we have w = 3. The dimensions that give the rectangle an area of 27 m2 are 9 m by 3 m.
Ways to Solve Quadratic Equations and When to Use:
Solving by Factoring
Example #1

Example #2
x  4 x  12  0
2x  7x  3  0

x  6 x  2   0
2 x  1x  3  0

x60 x20
2x 1 0 x  3  0
2
x6
x2
2
x
1
2
x3

Some quadratic equations can be solved by factoring. Set the equation
equal to zero and factor.
Solving by Graphing
Example #1 x 2  12 x   27
x  3 and x  9
Example #2 2 x 2  x  6
x 1.5 and x  2




Set the equation equal to zero, if
necessary. Find the roots using the
ZERO command tool of the graphing
calculator.
Graph each side of the equation
separately. Use the INTERSECT
command tool to find when the
graphs cross at both points.
Use When. . .
You are told to solve by
factoring.
The quadratic is easily
factorable.
The quadratic is already
factored, such as:
( x  2)( x  6)  0
The discriminant is positive and
a perfect square.
Use When. . .
You are told to solve by
graphing.
The graph easily shows integer
values for the x-intercepts.
An approximate solution is
sufficient.
The discriminant is positive
and/or = 0.
Solving by Square Roots
Example #1
Example #2
2 x  12  3
2
3 x  7   2  22
2x 1  3
2
3 x  7   24
 1 3
2x   1 3  x 
2
Use When. . .
 You are told to solve by the
square root method.
 x2 is set equal to a numeric
value, such as:
2
x  16
x  7 2  8
x  7   8  x  7  2 2
When using this method, you will have to make sure students use the “±” which ties
back to the quadratic formula and why there should be two answers.
 The middle term, bx, is
missing, such as:
2
3 x  15  0
You have the difference of two
squares, such as:
2
x  49  0
 The discriminant is positive
and/or = 0.
Solving by Quadratic
Formula
2
Example #1 3 x  5  0
a =3, b = 0, c = -5
Use When. . .
2
Example #2 9 x  6 x  4  0
a = 9, b = -6, c = -4
2
x
2
 b  b  4ac
x
 b  b  4ac
2a
x
x
23
 60
 x
x
 2 15
6
x
2
2a
  43 5
6
 15
3
x
x
6

10 x  3 x  4  0
 62  49  4 
29 
6  180
18
6 1 5
18
 You are told to use the
quadratic formula.
 Factoring looks difficult, or you
are having trouble finding the
correct factors.
 x

66 5
18
x
1 5
6
 The quadratic is not factorable,
such as:
2
3x  5  0
 The question asks for the
answer to be rounded.
 Solving any quadratic equation.
The solutions of some quadratic equations are not rational numbers and cannot be
factored. The equation should be set equal to zero before using the formula.
Solving by Completing
the Square
Example #1
2
x  8 x  25
2
Use When. . .
2
3 x  6 x  15  0
x  8 x  16  25  16
3 x  6 x  15
x  4 2  41
x  2x  5
x  4   41
x  4  41
2
a x  h   k  0
2
 a is 1 and b is even
 The discriminant is positive
and/or = 0.
2
2
x  2x  1  5 1
x  12  6
 x 1  6
x   1 6
When completing the square to solve an equation, be sure to add
coefficient of the x2 term is 1.
 You are told to solve by
completing the square.
 Have to put the quadratic in
vertex form, such as:
Example #2
b
 
2
2
to each side. Before you complete the square, be sure that the