# Algebra 1 Final Review

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```					                 Algebra 1 Final Review
1) What is the greatest common factor of
2x2y - 4xy3
• A factor is a number or variable    • Change the problem into 2xxy - 4
that divides into another number.     xyyy. You can divide both terms
• A common factor is a factor that      by 2xy.
divides into all terms.             • The problem changes into 2xy(x -
• The greatest common factor is the     2y2).
largest number or variable that     • The greatest common factor is
divides into all terms.               2xy.
2) The solution of the quadratic equation
x2 + 3x -5 =0 is:
• Remember that to solve a quadratic           3  32  (4  1 5)
x
equation you use the quadratic                      2 1
formula                                          3  9  ( 20)
x
b  b2  (4ac )
2
x                                            3  29
2a                                 x
• Remember also that in                                2
y

x2 + 3x -5 =0, a=1, b=3, and c=-5.                        2

x

• Substitute the values.                   -6    -4   -2

-2
0   2   4   6

-4

-6

-8
3) The difference of   3 xand   x
is:
4       2
• Remember that to subtract fractions,
3 x 2x x
you need common denominators.                   
4   4   4
• Multiply the second fraction by
.

2
2
x 2 2x
 
2 2  4
4) Find the roots of x2 - 4x + 3

• Remember that to solve a quadratic          (4)  42  (4  1 3)
equation you use the quadratic           x
2 1
formula
4  16  (12)
b  b  (4ac )
2                          x
x                                             2
2a
• Remember also that in                       4 4
x
x2 - 4x +3 =0, a=1,                           2
42
b=-4, and c=3.                           x      3 or
2
• Substitute the values.                        42
x     1
2
5) Which of the following is a factor of
x2 + 12x + 36?
equation, use the following strategy:   numbers that multiply to make 36 are
• You need to find two numbers that       1 and 36, 2 and 18, 3 and 12, 4 and 9,
add to make the x-term, but             and 6 and 6.
multiply to make the last term.
•The two numbers that add to make
• In x2 + 12x + 36, find two numbers      12 are 6 and 6.
that add to make 12, but also
•Write the factors as
multoply to make 36.
(x + 6)(x + 6) = 0.
6) What is the solution for x in the -y = 2x - 3 following system
of equations?         y = -x + 1
• Remember that the goal for systems -y = 2x - 3
of equations is to make one of the + y = -x + 1 Add the terms
variables add to make 0.             0=x-2
• Just add the equations here, because +2      +2 Solve for x
y + -y = 0.                          2=x
• -y = 2x - 3                           x=2                            y

4
+ y = -x + 1 Add the terms
2

0=x-2                                                                              x
-6   -4   -2        0   2   4   6

-2

-4
7) (-2x + 3) - (x - 4) =

• Remember that when you subtract a
quantity in parenthesis, the signs
change on the numbers.
• - ( x - 4) becomes - x + 4.
• Substitute and solve.
• -2x + 3 - x + 4
• -3x + 7
8) The solutions to the equation -x2 + 3x + 4 are:
• Remember that the solutions are the y-
intercepts, also called the roots. Use the        (3)  32  (4  1 4)
x
• First, multiply the entire equation by -1 to      3  9  ( 16)
make x2 positive. -1( -x2 + 3x + 4 ) is        x
2
x2 - 3x - 4 .
3  25
• In x2 - 3x - 4 , a = 1, b=-3, and c = -4.       x
2
Substitute in
35
x
2
35
x     4        or
b  b  (4ac )
2
2
x
2a                                        35
x        1
2
9) (3x2 - 4x + 1) + (-x2 +2x - 3) =

• Remember to add the like terms.
• (3x2 - 4x + 1) + (-x2 +2x - 3) becomes
• 3x2 - x2 -4x + 2x + 1 - 3
• 2x2 -2x -2
10) The width of a rectangle is x and the length is x + 3. What
expression finds the area?
• Remember that the area for the formula of a rectangle
is length times width, or A = l * w
• Substitute into the equation.
• A = (x + 3)*x
• Distribute. x * x = x2
• A = x2 + 3x
11) Where does the graph of                          x 2 5  14
f ( x )  crossthexx-axis?
b  b2  (4ac )
• Remember that roots or solutions are      x
2a
where the graph crosses the x-axis.
5  52  (4  1 14)
• Either factor or use the quadratic        x
equation to solve.                                   2 1
• Factor: find 2 numbers that add to make       5  25  56
x
5 and multiply to make -14.                        2
• The numbers are 7 and -2.                        5  81
• The factors are (x + 7)(x - 2)              x
2
• The solutions are what makes each              5  9
parenthesis 0, which are -7 and 2.          x         2 or
2
5  9
x         7
2
12) How many solutions does x2 + 2x + 5 have?

• Remember to use the discriminant         •   Use b2 - (4ac)
to find the number of solutions.         •   a =1, b=2, c=5
• If the discriminant is positive, there   •   22 - (4*1*5)
are 2 solutions; if 0, 1 solution; if    •   4 - 20
negative, no solutions.
•   -16
•   Since the answer is negative, there
are no solutions.
13) In 2x2 + 5 = 11x, what are the values for a, b, and c?

• Remember that the quadratic                                  y
x
equation is
-6   -4   -2         0   2   4   6
ax2 + bx + x = 0.
• You need to subtract 11x from                           -5
both sides of the equation to fit the
equation.
• 2x2 + 5 = 11x                                          -10

-11x -11x
2x2 -11x + 5 = 0
-15
• a=2, b=-11, c=5
14) What is the graph of                  f ( x )  2 x 2  x  6?
•   Look for 4 things to see if a graph is the correct
one:                                                     • This graph has
•   The line of symmetry is the center of the graph.            -1/4 as a line of
Use                          (
b       1
)  ( )
symmetry and a y-
2a       4                  intercept of -6
•   The vertex is the highest or lowest point of the
graph. Find the line of symmetry, then substitute                              y

the                                                                        5

x
•    The y-intercept is the last number in a quadratic         -6   -4   -2        0   2   4   6

equation, or -6.
-5
•   The solutions or roots is where the graph crosses
the x-axis.
15) What is the graph of              f ( x )  x 2  2 x  3?
• Look for 4 things to see if a graph is the correct
one:                                                      • This graph has
• The line of symmetry is the center of the graph.            -1 as a line of symmetry
Use                         (
b       2
)  ( )  1
and a y-intercept of -3
2a       2
• The vertex is the highest or lowest point of the
graph. Find the line of symmetry, then substitute                          y
the
5
• The y-intercept is the last number in a quadratic
equation, or -3.                                     -6     -4   -2        0   2   4   6
x

• The solutions or roots is where the graph crosses
the x-axis.
-5
16)       x6          5x
           
x 2 x 2
• Treat both of these as regular fractions. To
divide, multiply by the inverse or reciprocal
of the last number.

x6        5x        x 6 x 2
                 
x  2 the 2
• Cancel outx  x - 2. x  2 5 x

x 6 x 2 x 6
   
x  2 5x   5x
17) Complete the square for x2 +6x -2 = 0.
• To complete the square, get rid of the
number after the x-value, or -2.                                 y
• x2 +6x -2 = 0                                                            x
-6   -4   -2         0   2
+2 +2
x2 +6x = 2
• Take half of the x-value, and use it as
factors. Then square it and add it to the                   -5
right side of the equation.
• (x+3)(x+3)=2+32
• (x+3)(x+3)=11       Simplify
• (x+3)2 = 11                                                -10
18) What are the coordinates of the vertex for for x2 +2x -15 = 0.

• Find the line of symmetry and the x-value of
the equation using                                                  y               x
b          2                -6   -4   -2         0   2   4   6
x  ( )  ( )  1
2a
• Substitute this x-value in the2quadratic
equation to find the y-value.                                -5

•   x2 +2x -15 = y
•   (-1)2 +2(-1) -15 = y                                        -10
•   1-2-15 = y
•   y = -16
-15
•   The vertex is (-1, -16)
19) What are the solutions of (3x - 5) (x + 7) =0?

• Use the factored form of the quadratic                                        x
y
equation to find the roots or solutions.         -6   -4   -2         0   2
• In (3x - 5) (x + 7) =0, either (3x- 5)=0 or (x
+7)=0. Solve for each value to find the
solutions.
-20
• 3x - 5 = 0                  x+7=0
+5 +5                      -7 -7
3x = 5                    x = -7
3 3                                                             -40
x=5
3
20) If ab = 0, what must be true?

• Remember that any number times 0 is 0.
• Either a is 0, or b is 0

a0  0
b0  0
21) Which graph of a quadratic function has -2 as a root?

• Remember that a quadratic equation is a
parabola. It always has an x2 value.
• Remember that a root is a solution, or x-
intercept.
• Look for a graph of a parabola with -2 as an
x-intercept.
22) Simplify                2 x  10 factoring.
by
x  2 x  15
2

• Remember that factoring means dividing out
a common number.                                            2( x  5)
• In the numerator, divide a 2 out of both                ( x  5)( x  3)
terms.                                           •Cancel out the x + 5
2( x  5)
2
x 2  2 x  15                                         ( x  3)
• In the denominator, factor to find 2
numbers that add to make the x-value, -2,
but also multiply to make the last value, -15.
• The factors are 3 and -5. The denominator
is (x + 3)(x - 5).
23) What are the factored forms and solutions of
2x 2  x  3  0
• Work in reverse on this one. Multiply out
the answers to match the question. Use
FOIL to expand the factors.
• (x-1)(2x+3) =2x2 +x -3. These are the
correct factors. Now find the solutions by
setting each factor to 0.
• x-1=0, so x=1. 2x+3=0, so x=-3/2.
24) What quadratic equation is equivalent to
(2x + 4)(x - 6)?                   •FOIL Method
(2x + 4)(x - 6)
• You either need to use FOIL or the vertical
method to multiply the quantities.            •Multiply the First numbers
• Vertical Method:                              2x*x=2x2
2x + 4
•Multiply the Outside numbers
* x - 6
-12x - 24                        2x* -6=-12x
2x2 + 4x
•Multiply the Inside numbers
2x2 - 8x - 24
4*x=4x
•Multiply the Last numbers
4*-6=-24
•Add together 2x2 - 8x - 24
25) Find the x-intercepts of x2 + 3x + 2 = 0.
• You can either use factoring or the quadratic
formula to find the x-intercepts or solutions.   •a=1, b=3, c=2
• Factoring:                                            b  b2  (4ac )
Find 2 numbers that add to make the                x
2a
middle number, 3, but also multiply to make
the last number, 2.                                   3  32  (4  1 2)
x
• The numbers that work are 1 and 2. Put                      2 1
them in factors.                                       3  9  8
x
• (x+1)(x+2) =0.                                              2
• Set each factor to 0. If x+1 = 0, then x = -1.          3  1      or
If x+2 = 0, then x = -2. Solutions are -1 and       x          1
2
-2.                                                      3  1
x         2
2
26) Which area of the graph shows the solutions to the system of
inequalities y > 2x+1, y > -x - 1 ?
•Use a point not on each inequality line as      •0 > -0 -1
a test case. I usually use the point (0,0).      •0 > -1. This is true, so shade the side of
•y > 2x + 1 Substitute (0,0).                    the line where (0,0) is. Shade vertically.

•0 > 2*0 + 1                                     •The answer is where the horizontal and
•0 > 1
y
•Since this is false, shade in the side of the                         Area A
line where (0,0) isn’t. Shade horizontally.                                       4

•Do the same thing for                                                            2

x
•y > -x - 1                                                      -6   -4    -2        0   2

-2

-4
27) Which area of the graph is the solution to the system of
inequalities y > 3x, y < 6 - x?
•Use a point not on each inequality line as     •2 < 6 - 0
a test case. Use the point (0,2).
•2 < 6. This is true, so shade the side of
•y > 3x Substitute (0,2).                       the line where (0,2) is. Shade vertically.
•2 > 3*0                                        •The answer is where the horizontal and
•2 > 0                                          vertical shading meet.
•Since this is true, shade in the side of the
y
line where (0,2) is. Shade horizontally.
6
•Do the same thing for                                       Area D
4
•y < 6 - x
2

x
-2         0   2   4   6
28) The solution set of x2 - 36 = 0 is:
•This is called the difference of squares
pattern. The general pattern is to take the
square root of both numbers and place
them in the pattern (a+b)(a-b).

•In this case, take the square root of x2 and
36, which are x and 6. Place them in the
pattern (x+6)(x-6) = 0.
•The solutions are what makes each
quantity 0. For x+6,
x-= -6; for x-6, x = 6. {6, -6}
29) To find the roots of a quadratic equation in standard form,
ax2+bx+c=0, use x=
•The roots are also the solutions. To find
the solutions, use the quadratic formula.

b  b2  (4ac )
x
2a
30) Simplify the expression                 x  6t 5t  6

3t     3t
•To subtract fractions, you need common
denominators. Since the denominators are    8  6t  5t  6
already the same, you can combine the             3t
fractions.
t  14
3t
8  6t  (5t  6)
3
•Distribute the -tsign, then combine like
terms.
31) Simplify            b5
b2
•Numbers with exponents can be reduced if
they have the same bases. Since the base is
b in both terms, you can combine the
exponents.
•Get rid of the dividing number by
subtracting the bottom exponent from the
top exponent.

b5
 b52  b3
b2
32) What is the solution of the equations graphed below?
y                    The solution to the equations is where
the two graphs cross, at (4,3).
6

4

2

x
-2        0   2   4   6

-2
33) Find the sum of                    5 x 5

2x   2x
•You can add fractions when the
denominators are the same. Since the
denominators are the same, add the
numerators.

5 x 5
2x
1x
2x
2
1
34) What are the roots of the equation
3x2+6x - 2=0?
•The roots are the solutions, or x-            6  36  ( 24)
intercepts. Use either factoring or the   x
6
6  60
•a=3, b=6, c=-2                           x
6

b  b2  (4ac )
x
2a

6  62  (4  3  2)
x
23
35) Find the solution to x2 - 7x+10=0.
•The roots are the solutions, or x-intercepts.
Use either factoring or the quadratic formula.
•a=1, b=-7, c=10
•Factoring: Find 2 numbers that add to make
-7 but also multiply to make 10.
•The numbers are -2 and -5
•Make the factors (x-2)(x-5)=0
•Solve each factor for 0. For x-2=0, x is 2; for
x-5=0, x is 5.
36) What is equivalent to (73)2?
•When you raise an exponent to another
exponent, multiply the exponents.
•(73)2 =76
37) Simplify the expression                      15 x 3 y 2
3 xy
•The easiest way is to expand the terms, then   •A second way is to use the exponents.
reduce.

15 xxxyy
3 xy                                       15 x 3 y 2
 5 x 3 1y 21
3 xy
5xxy
y2 x5                                        5x 2 y
38) Simplify           x 2  7 x  12
x2  9
•Factor the top and bottom terms.             •This is the difference of squares pattern.
x 2 2 7x  that add to make 7, but also
•Find numbers 12                               Put  3)( x  3) together and cancel
( x the problem back
• possible.
multiply to make 12.                          where

•The numbers are 3 and 4. Yhe factors are
(x+3)(x=4)                                       ( x  3)( x  4)
•Find the factors of                             ( x  3)( x  3)

•Take the square root of both terms and put       ( x  4)
them into factors.            x2  9              ( x  3)
39) Simplify            55
53
•Since the numbers have the same base, you
can subtract the exponents.

55
 5 6  3  53
53
40) The greatest common factor of 12x2 and 8x3 are:
•Remember that factors are numbers that         •The greatest common factors are 4 and x2, or
divide into others. Common factors are          4x2.
numbers or variables that divide into both
terms. Greatest common factors are the
largest numbers or variables that divide into
both terms.
•The number that divides into both 12 and 8
is 4. The variable that divide sinto both x2
and x3 are x2.
41) Simplify 32 * 35.
•When you’re multiplying the same bases with
•32 * 35 =32+5 = 37
42) Find the solution to y=3x-4 and 6x-2y=8.
•First, put both equations in standard form,   •2(-3x + y = -4) Multiply by 2
with both the x’s and y’ on the same side.
•-6x +2y = -8
• y=3x-4
•-3x -3x
•-6x +2y = -8
•-3x + y = -4
• +6x - 2y = 8
• 6x - 2y= 8
•      0=0
•The goal is to get either the x’s or the y’s to
become 0 by multiplying the entire equation •In this case, there are an infinite number of
times some number.                               solutions.
•(If it was x=7, there woulld be one solution, or
6=7 no solutions.
43) What is .00437 in scientific notation?
•Scientific noation states that you can only
have one number to the left of the decimal
point.
•The decimal point has to be between the 4
and the 3.
•The decimal point had to move 3 places, and
because the number is less than 1, the
exponent is negative.
•4.37 * 10-3
45) The product of                  2a 3 is:3a 2
(      )(      )
5b 7b
•Multiply the numerators, then multiply the
denominators.
2  a3  3  a 2
5b7b

6  a3  2
35  b 2
6a 5
35b 2
46) What is the square root of 16?
•What number times itself = 16?
•4*4=16
•-4*-4=16
•The square roots are 4 and -4
49) When 8x4 - 8x is divided by 8x, the quotient is:
•Divide each term by 8x.

8x 4 8x

8x 8x
x 41  1
x 1
3
50) Substitute 2x2 +3x +4 = 0 into the quadratic formula. Do not solve.
•a=2, b=3, c=4

b  b2  (4ac )
0
2a
3  32  (4  2  4)
0
22

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