Deck 6: Factoring and Quadratic Equations

Question 1

Factor by grouping. $$2 a ^ { 2 } - 3 b c - 2 a b + 3 a c$$

A) $2 a + 3 c ( a - b )$
B) $( 2 a - 3 c ) ( a + b )$
C) $( 2 a + 3 c ) ( a - b )$
D) $( a + c ) ( a - b )$
E) $2 a ^ { 2 } - 2 a b + 3 c ( a - b )$

Answer 1

$( 2 a + 3 c ) ( a - b )$

Question 2

Factor the trinomial. Factor out any common monomials first (including $-$ 1 if necessary). $$- 7 u ^ { 2 } + 42 u - 63$$

A)7 ( u $-$ 3)( u $-$ 3)
B) $-$ 7 ( u + 3)( u $-$ 3)
C)7 ( u $-$ 3)( u + 3)
D) $-$ 7 ( u $-$ 3)( u $-$ 3)
E) $-$ 7 ( u $-$ 3)( u + 3)

Answer 2

$-$ 7 ( u $-$ 3)( u $-$ 3)

Question 3

Factor the trinomial. Factor out $-$ 1 first. $$- a ^ { 2 } + 11 a - 30$$

A)( a $-$ 5)( a $-$ 6)
B) $-$ ( a $-$ 5)( a $-$ 6)
C) $-$ ( a + 5)( a $-$ 6)
D)( a + 5)( a $-$ 6)
E) $-$ ( a $-$ 5)( a + 6)

Answer 3

$-$ ( a $-$ 5)( a $-$ 6)

Question 4

Factor the trinomial, if possible. $$12 z ^ { 2 } + 31 z + 20$$

A) $( 3 z + 4 ) ( 5 z - 5 )$
B) $( 3 z + 4 ) ( 2 z + 5 )$
C) $( 3 z + 4 ) ( 4 z + 5 )$
D) $( 3 z + 4 ) ( 3 z - 5 )$
E)prime

Question 5

Factor the following perfect square trinomial. $$a ^ { 2 } + 6 a + 9$$

A) $( a - 3 ) ^ { 2 }$
B) $( a + 2 ) ^ { 2 }$
C) $( a + 6 ) ^ { 2 }$
D) $( a + 3 ) ^ { 2 }$
E)none of these

Question 6

Factor completely. $$91 x ^ { 3 } y ^ { 4 } - 77 x ^ { 4 } y ^ { 3 }$$

A) $7 x ^ { 3 } y ^ { 3 } ( 13 y - 11 x )$
B) $7 y ^ { 3 } \left( 13 x ^ { 3 } y - 11 x ^ { 4 } \right)$
C) $7 x ^ { 3 } \left( 13 y ^ { 4 } - 11 x y ^ { 3 } \right)$
D) $7 \left( 13 y ^ { 4 } y - 11 x ^ { 4 } \right)$
E) $x ^ { 3 } y ^ { 3 } ( 91 y - 77 x )$

Question 7

Factor by grouping. $$a x ^ { 4 } - x ^ { 4 } + 2 a - 2$$

A) $a x ^ { 4 } + 2 a - \left( x ^ { 4 } + 2 \right)$
B) $\left( x ^ { 4 } + 2 \right) a - x ^ { 4 } - 2$
C) $\left( x ^ { 4 } + 2 \right) ( a - 1 )$
D) $x ^ { 4 } + 2 ( a - 1 )$
E) $\left( x ^ { 4 } - 2 \right) a + x ^ { 4 } + 2$

Question 8

Factor the trinomial. $$y ^ { 2 } + 16 y + 63$$

A) $( y + 7 ) ( y + 9 )$
B) $( y + 7 ) ( y + 5 )$
C) $( y + 7 ) ( y - 5 )$
D) $( y + 7 ) ( y - 7 )$
E) $( y + 7 ) ( y - 9 )$

Question 9

Find the greatest common factor of the numbers below. 111, 123 and 51

A)4
B)3
C)2
D)5
E)none of these

Question 10

Find the GCF of 40, 45, and 50.

A)10
B)8
C)1
D)5

Question 11

Factor the trinomial. $x ^ { 2 } - 18 x + 77$

A) $( x - 11 ) ( x + 13 )$
B) $( x - 7 ) ( x + 11 )$
C) $( x - 7 ) ( x + 11 )$ .
D) $( x - 9 ) ( x - 9 )$
E) $( x - 11 ) ( x + 9 )$

Question 12

Find the GCF of the numbers below. $12$ , $25$ and $63$

A)1
B)2
C)3
D)4

Question 13

Factor the expression. $$y ^ { 2 } - 4 y + 4$$

A) $( y + 2 ) ( y + 2 )$
B) $( y - 2 ) ( y + 2 )$
C) $( y + 4 ) ( y + 1 )$
D) $( y - 2 ) ( y - 2 )$
E)prime

Question 14

Factor completely. $6 x ^ { 2 } + 33 x$

A) $3 \left( 2 x ^ { 2 } + 11 x \right)$
B) $2 \left( 3 x ^ { 2 } \right) + 11 ( 3 x )$
C) $2 x ( 3 x ) + 11 ( 3 x )$
D) $x ( 6 x + 33 )$
E) $3 x ( 2 x + 11 )$

Question 15

Factor by grouping. $$13 x ^ { 2 } + 91 x - 3 x - 21$$

A) $( 13 x - 3 ) ( x + 7 )$
B) $13 x ( x + 7 ) - 3 ( x + 7 )$
C) $13 x ( x + 7 ) - 3 x + 21$
D) $( 13 x - 3 ) ( x - 21 )$
E) $13 x ^ { 2 } + 91 x - 3 ( x + 7 )$

Question 16

Factor the trinomial. Factor out the GCF first. $$24 x ^ { 6 } - 10 x ^ { 4 } - 21 x ^ { 2 }$$

A) $2 x ^ { 2 } \left( 6 x ^ { 2 } - 7 \right) \left( x ^ { 2 } + 3 \right)$
B) $\left( 6 x ^ { 2 } - 7 \right) \left( 4 x ^ { 2 } + 3 \right)$
C) $x ^ { 2 } \left( 6 x ^ { 2 } - 8 \right) \left( 4 x ^ { 2 } - 3 \right)$
D) $x ^ { 2 } \left( 6 x ^ { 2 } - 5 \right) \left( 4 x ^ { 2 } - 3 \right)$
E) $x ^ { 2 } \left( 6 x ^ { 2 } - 7 \right) \left( 4 x ^ { 2 } + 3 \right)$

Question 17

Factor the trinomial completely. $$8 x ^ { 2 } + 4 x - 24$$

A)( x - 2)(8 x - 12)
B)8 ( x + 1)( x - 3)
C)4 ( x + 2)(2 x - 3)
D)(4 x + 8)(2 x - 3)
E)not factorable

Question 18

Factor completely. $$8 x ^ { 2 } - 13 x ^ { 4 } + 12 x ^ { 5 }$$

A) $8 x ^ { 2 } - x ^ { 4 } ( 13 + 12 x )$
B) $x ^ { 2 } \left( 8 - 13 x ^ { 2 } \right) + 12 x ^ { 5 }$
C) $x \left( 8 - 13 x ^ { 3 } + 12 x ^ { 4 } \right)$
D) $x ^ { 2 } \left( 8 + 12 x ^ { 3 } \right) - 13 x ^ { 4 }$
E) $x ^ { 2 } \left( 8 - 13 x ^ { 2 } + 12 x ^ { 3 } \right)$

Question 19

Factor the expression. $$x ^ { 2 } + 7 x + 3$$

A) $( 3 x + 1 ) ( x + 1 )$
B) $( 2 x + 1 ) ( x + 1 )$
C) $( 2 x + 3 ) ( x - 1 )$
D) $( 2 x - 1 ) ( x + 2 )$
E)prime

Question 20

Factor the trinomial completely. $$2 x ^ { 2 } + 9 x + 4$$

A) $( 2 x + 1 ) ( x + 4 )$
B) $( 2 x - 1 ) ( x - 4 )$
C) $( 2 x + 2 ) ( x + 4 )$
D) $( x + 1 ) ( 2 x + 4 )$
E)not factorable

Question 21

Factor the trinomial, if possible. $$8 a ^ { 2 } + 21 a - 9$$

A) $( 8 a + 3 ) ( a - 3 )$
B) $( 8 a + 3 ) ( a - 4 )$
C) $( 8 a + 3 ) ( a - 5 )$
D) $( 8 a - 3 ) ( a + 3 )$
E)prime

Question 22

Factor the polynomial. $$9 z ^ { 2 } - 78 z + 169$$

A) $3 ( z + 13 ) ^ { 2 }$
B) $( 3 z + 13 ) ^ { 2 }$
C) $3 ( 13 - z ) ^ { 2 }$
D) $( 3 z - 13 ) ^ { 2 }$
E) $3 ( z - 13 ) ^ { 2 }$

Question 23

Factor the trinomial. Factor out the GCF first. $$2 x ^ { 2 } + 8 x - 64$$

A) $( x + 8 ) ( x - 4 )$
B) $2 ( x + 11 ) ( x - 4 )$
C) $2 ( x + 8 ) ( x - 4 )$
D) $( x + 7 ) ( x - 4 )$
E) $2 x ( x - 8 ) ( x - 7 )$

Question 24

Factor the following perfect square trinomial. $$x ^ { 2 } + 18 x + 81$$

A) $( x - 9 ) ^ { 2 }$
B) $( x + 8 ) ^ { 2 }$
C) $( x + 18 ) ^ { 2 }$
D) $( x + 9 ) ^ { 2 }$
E)none of these

Question 25

Factor the expression, if possible. Factor out any GCF first (including -1 if the leading coefficient is negative). $$5 - x ^ { 2 } - 4 x$$

A) $( x + 6 ) ( x + 1 )$
B) $- ( x + 5 ) ( x - 1 )$
C) $- ( x + 2 ) ( x + 1 )$
D) $( x + 3 ) ( x - 1 )$
E) $- ( x - 4 ) ( x + 1 )$

Question 26

Factor the polynomial. $$9 - y ^ { 2 } c ^ { 2 }$$

A) $( 3 + y c ) ( 3 - y c )$
B) $( 3 - y c ) ( 3 - y c )$
C) $( 3 + y c ) ( 3 + y c )$
D) $( 2 - y c ) ( 3 - y c )$
E) $( 2 + y c ) ( 3 - y c )$

Question 27

Factor the expression, if possible. Factor out any GCF first (including -1 if the leading coefficient is negative). $$4 x ^ { 2 } - 4 x + 1$$

A) $( 2 x + 1 ) ^ { 2 }$
B) $( 2 x - 2 ) ^ { 2 }$
C) $( 2 x - 1 ) ( 2 x + 1 )$
D) $( 2 x - 1 ) ^ { 2 }$
E)prime

Question 28

Factor the following perfect square trinomial. $$49 x ^ { 2 } + 14 x + 1$$

A) $( 7 x + 14 ) ^ { 2 }$
B) $( x + 7 ) ^ { 2 }$
C) $( 7 x - 1 ) ^ { 2 }$
D) $( x - 7 ) ^ { 2 }$
E) $( 7 x + 1 ) ^ { 2 }$

Question 29

Factor the polynomial. $$z ^ { 2 } + 6 z + 9$$

A) $( z - 3 ) ^ { 2 }$
B) $( z + 3 ) ^ { 2 }$
C) $( z + 3 ) ( z - 3 )$
D) $( z + 3 ) ( z + 3 )$
E)none of these

Question 30

Factor the expression. $$( a - m ) c + ( a - m ) f$$

A) $( f - m ) ( c + a )$
B) $( a - f ) ( c + m )$
C) $c ( a - f ) + f$
D) $( a - m ) ( c + f )$
E) $( m - a ) ( c + f )$

Question 31

Factor the expression. Factor out any greatest common factors first. $$4 x ^ { 3 } + 32$$

A) $4 ( x - 2 ) \left( x ^ { 2 } + 8 x + 4 \right)$
B) $4 ( x + 4 ) \left( x ^ { 2 } + 4 x + 4 \right)$
C) $2 ( x - 2 ) \left( x ^ { 2 } + 2 x + 4 \right)$
D) $4 ( x + 2 ) \left( x ^ { 2 } - 2 x + 4 \right)$
E) $2 ( x + 4 ) \left( x ^ { 2 } + 4 x - 16 \right)$

Question 32

Factor the polynomial. $$y ^ { 2 } + 18 y + 81$$

A) $( y + 9 ) ^ { 2 }$
B) $( 9 + y ) ( 8 - y )$
C) $( y - 9 ) ( y + 9 )$
D) $( y - 9 ) ^ { 2 }$
E)none of these

Question 33

Factor the polynomial completely. $64 - 25 y ^ { 2 }$

A)(8 + y )(8 + 5 y )
B)(8 - 5 y )(8 + 5 y )
C)(64 - 5 y )(64 + 5 y )
D)(8 - 25 y )(8 + 25 y )
E)none of these

Question 34

Factor the following perfect square trinomial. $$25 a ^ { 2 } - 60 a y + 36 y ^ { 2 }$$

A) $( 5 y - 6 a ) ^ { 2 }$
B) $( 5 a - 6 y ) ^ { 2 }$
C) $( 6 a + 5 y ) ^ { 2 }$
D) $( 6 a - 5 y ) ^ { 2 }$
E) $( 5 a + 6 y ) ^ { 2 }$

Question 35

Use the difference of squares pattern to factor the polynomial. $$x ^ { 2 } - 9 y ^ { 2 }$$

A)( x - 9 y )( x + 9 y )
B)( x + 3 y )( x + 9 y )
C)( x - 3 y )( x + 3 y )
D)( x - 3 y )( x - 9 y )
E)none of these

Question 36

Factor the expression. $$y ^ { 3 } + 64$$

A) $( y + 4 ) ^ { 3 }$
B) $( y - 4 ) \left( y ^ { 2 } + 4 y - 16 \right)$
C) $( y + 4 ) \left( y ^ { 2 } - 4 y + 16 \right)$
D) $( y - 4 ) \left( y ^ { 2 } - 4 y - 16 \right)$
E) $( y - 4 ) ^ { 3 }$

Question 37

Give the name for this type of equation. 6 x ² + 9 x + 8 = 0

A)cubic
B)quadratic
C)linear
D)true
E)none of these

Question 38

Factor the polynomial. $$8 a ^ { 3 } - 27 b ^ { 3 }$$

A) $( 2 a - 3 b ) \left( 2 a ^ { 2 } + 6 a b + 3 b ^ { 2 } \right)$
B) $( 2 a + 3 b ) \left( 2 a ^ { 2 } + 6 a b + 3 b ^ { 2 } \right)$
C) $( a - b ) \left( 4 a ^ { 2 } + 6 a b + 9 b ^ { 2 } \right)$
D) $( a - b ) \left( 4 a ^ { 2 } + 6 a b - 9 b ^ { 2 } \right)$
E) $( 2 a - 3 b ) \left( 4 a ^ { 2 } + 6 a b + 9 b ^ { 2 } \right)$

Question 39

Factor the expression. $$- 30 a b c - 30 a c ^ { 2 } + 5 b c + 5 c ^ { 2 }$$

A) $- 5 c ( b - c ) ( 6 a + 1 )$
B) $- 5 c ( b - 1 ) ( 6 a + c )$
C) $- 5 c ( b + c ) ( 6 a - 1 )$
D) $5 c ( b - 1 ) ( 6 a + c )$
E) $- 5 c ( b + 1 ) ( 6 a - c )$

Question 40

Factor the expression. $$( x - 4 ) ^ { 2 } + ( x - 4 )$$

A) $( x - 4 ) ( x + 3 )$
B) $( x - 6 ) ( x - 3 )$
C) $( x - 4 ) ( x - 5 )$
D) $( x - 4 ) ( x + 5 )$
E) $( x - 4 ) ( x - 3 )$

Question 41

Paul is elder than his brother Bryan. Their ages are consecutive integers whose product is 552. How old are Bryan and Paul?

A)Bryan is 23, Paul is 24
B)Bryan is 19, Paul is 20
C)Bryan is 27, Paul is 28
D)Bryan is 31, Paul is 32

Question 42

Solve the equation. $x ^ { 3 } - 11 x ^ { 2 } + 30 x = 0$

A)1, 6 , 5
B)0, 6, 5
C)6 , 5
D)-6, -5
E)0, -6 , -5

Question 43

The equation $7 x ^ { 2 } - 5 x - 1 = 0$ is written in $a x ^ { 2 } + b x + c = 0$ form. What are a, b and c ?

A)a = $7$ , b = $5$ , c = $1$
B)a = $7$ , b = $- 5$ , c = $- 1$
C)a = $- 7$ , b = 0, c = $1$
D)a = $- 5$ , b = $7$ , c = $- 1$

Question 44

To write a quadratic equation in standard form means that one side of the equation must be zero and the other side must be in the form:

A) $a x + b$
B) $a ^ { 2 } + b x + c$
C) $a x ^ { 2 } + b x + c$
D) $x ^ { 2 } + a c$

Question 45

Solve the equation. $x ^ { 3 } + 15 x ^ { 2 } = x ^ { 2 } - 49 x$

A) $x = 7,0$
B) $x = - 7$
C) $x = - 7,0$
D) $x = - 7,7,0$
E) $x = 7$

Question 46

A rectangle has an area 390 square centimeters. Its length is 4 centimeters more than twice its width. Find the perimeter of the rectangle.

A)78
B)98
C)94
D)86

Question 47

When a wholesaler sells n CD players, the revenue is $R = 150 n - \frac { 1 } { 2 } n ^ { 2 }$ . How many CD players would the wholesaler have to sell to receive $\$ 11,250$ ?

A) $120$ CD players
B) $190$ CD players
C) $150$ CD players
D) $130$ CD players
E) $140$ CD players

Question 48

The inclined ramp of the boat launch shown in the illustration is 10 meters longer than the "rise" of the ramp. The "run" is 5 meters longer than the "rise". How long is the "rise" of the ramp?

A)25 m
B)7.5 m
C)20 m
D)15 m
E)14 m

Question 49

Factor the trinomial square and use the square root method to solve the equation. $3 x ^ { 2 } + 7 x + 1 = 2$

A) $\frac { ( - 10 + \sqrt { 61 } ) } { 7 }$ , $\frac { ( - 10 - \sqrt { 61 } ) } { 7 }$
B) $\frac { ( - 6 + \sqrt { 61 } ) } { 7 }$ , $\frac { ( - 6 - \sqrt { 61 } ) } { 7 }$
C) $\frac { ( - 7 + \sqrt { 61 } ) } { 6 }$ , $\frac { ( - 7 - \sqrt { 61 } ) } { 6 }$
D) $\frac { ( - 8 \sqrt { 61 } ) } { 7 }$ , $\frac { ( - 8 - \sqrt { 61 } ) } { 7 }$

Question 50

Solve the equation. ( x - 5 )( x + 1 )= 0

A)5, -6
B)6, -5
C)5, -1
D)-5, 1
E)none of these

Question 51

Solve the equation: $p ( 9 p + 2 ) = 0$

A) $0 , - \frac { 2 } { 9 }$
B) $0 , - 2$
C) $- 1,2$
D) $1 , - \frac { 2 } { 9 }$

Question 52

The product of two consecutive integers is $1406$ . Find the integers.

A) $25$ , $26$
B) $31$ , $32$
C) $37$ , $38$
D) $43$ , $44$

Question 53

The x - and y -coordinates of a point in quadrant III are consecutive even integers whose product is 360. Find the coordinates of the point.

A)(20, 18)
B)(-20, -18)
C)(-24, -22)
D)(24, 22)

Question 54

Factor the trinomial square and use the square root method to solve the equation. $x ^ { 2 } - 9 x + 1 = 1$

A) $0$
B) $\pm9$
C) $0 , - 9$
D) $0,9$
E) $9$

Question 55

Use the factoring method to solve the equation. $6 x ^ { 2 } + 13 x + 5 = 0$

A) $- \frac { 1 } { 2 }$ , $- \frac { 5 } { 3 }$
B) $\frac { 5 } { 3 }$ , $\frac { 1 } { 2 }$
C) $- \frac { 5 } { 3 }$ , $\frac { 1 } { 2 }$
D) $- \frac { 1 } { 2 }$ , $\frac { 1 } { 2 }$
E) $- \frac { 1 } { 2 }$ , $\frac { 5 } { 3 }$

Question 56

A booklet is rectangular and its length is 2 centimeters less than 2 times its width. Its area is 40 square centimeters. Find the dimensions of the booklet.

A)5 cm by 8 cm
B)4 cm by 14 cm
C)4 cm by 8 cm
D)5 cm by 14 cm

Question 57

The distance (in feet)that an object will fall in t seconds is given by the formula $s = 16 t ^ { 2 }$ . How long will it take an object to fall 576 feet?

A)6 sec
B)36 sec
C)8 sec
D)7 sec
E)35 sec

Question 58

The height in feet h of a rocket in flight is given by the formula $h = - 16 t ^ { 2 } + 224 t$ , where t is the time of the flight in seconds. How long will it take for the rocket to hit the ground?

A)13 sec
B)14 sec
C)16 sec
D)12 sec
E)11 sec

Question 59

The length of the rectangular desk is twice its width. Its area is 0.98 square meters. Find the dimensions of the desk.

A)0.5 m by 1.2 m
B)0.7 m by 1.4 m
C)0.9 m by 1.6 m
D)1 m by 1.7 m

Question 60

Solve the equation: $( 5 y - 2 ) ( y + 7 ) = 0$

A) $- \frac { 2 } { 5 } , 7$
B) $- 7 , \frac { 2 } { 5 }$
C) $- 7,2$
D) $- 5 , \frac { 2 } { 7 }$

Question 61

The longer leg of a right triangle is 2 centimeters longer than twice its shorter leg. The hypotenuse is 1 centimeter longer than longer leg. Find the dimensions of the lengths of the triangle.

A)3 cm, 4 cm, 5 cm
B)2 cm, 5 cm, 10 cm
C)10 cm, 17 cm, 18 cm
D)5 cm, 12 cm, 13 cm

Question 62

The longer leg of a right triangle is 3 units longer than the shorter leg and 3 units less than hypotenuse. Find the length of the shorter leg of the triangle.

A)12
B)3
C)9
D)15