Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

GMAT > Algebra > Flashcards

Algebra Flashcards

1
Q

Definitions :
constant
term
coefficient

A

constant : a nb, or a symbol, such as Pi, that doesn’t change in value

term : a product of constants and variables, including powers of variables 
5
x
6y²
x²y²z²

coefficient : the constant factor of a term
in 6y², 6 is the coefficient
when no constant is written, the coefficient is 1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

(a+b)² =
(a-b)² =
a²-b² =

A 
(a+b)² = a² + 2ab + b²
(a-b)² = a² - 2ab + b²
a²-b² = (a+b)(a-b)
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q 
Factor the following expressions
x² - 49 =
9x² -16 =
25x² - 64y² =
x²y² - 1 =
A

x² - 49 = ( x + 7 )( x - 7 )
9x² -16 = ( 3x - 4 )( 3x + 4 )
25x² - 64y² = ( 5x + 8y )( 5x - 8y )
x²y² - 1 = ( xy - 1 )( xy + 1 )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q 
Factor the following expressions
x^6 - 16 = 
x^8 - 9y² =
x^7 - 4x^5 =
x^4 - 81 =
A

x^6 - 16 = ( x^3 - 4 ) ( x^3 + 4 )
x^8 - 9y² = ( x^4 + 3y ) ( x^4 - 9y)
x^7 - 4x^5 = x^5 ( x - 2 ) ( x + 2)
x^4 - 81 = ( x² + 9 ) ( x² - 9 ) = ( x² + 9 )( x - 3 )( x + 3 )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Factor the following expressions
y² - 25 =
25s² - t^4 =
x^9 - x =

A

y² - 25 = (y - 25)(y + 25)
25s² - t^4 = (5s - t²)(5s + t²)
x^9 - x = x(x^8 - 1) = x(x^4 - 1)(x^4 + 1) = x(x^4 + 1)(x² + 1)(x² - 1)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q 
If y = 5 + x 
and y = 12 - x
and if y² = x² + K
then K equals which of the following? 
A : 17
B : 25
C : 60
D : 119
A 
y = 5 + x   so   y - x = 5
y = 12 - x   so   y + x = 12

K = y² - x² = ( y - x )( y + x) = 5*12 = 60

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

(x + p)(x + q) =

A

x² + qx + px + pq
x² + (p + q)x + pq

we need to find the unknown constants p & q
the linear coefficient of the quadratic is the SUM of these two unknowns, and the constant term of the quadratic is the PRODUCT of the two unknowns

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q 
Factor :
x² + 8x +15
x² + 14x + 24
x² + 4x -21
x² - 2x - 35
x² - 16x + 48
A

x² + 8x + 15
we need to find two numbers whose sum is 8 and whose product is 15
x² + 8x + 15 = (x+3)(x+5)

x² + 14x + 24 = (x+2)(x+12)

x² + 4x -21
since the sum is positive, the larger nb must be positive
+7 and -3
x² + 4x -21 = (x+7)(x-3)

x² - 2x - 35
one negative, one positive, one with bigger absolute value is negative
+5 and -7
x² - 2x - 35 = (x-7)(x+5)

x² - 16x + 48
both negative
-12 and -4
x² - 16x + 48 = (x-12)(x-4)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q 
Factor :
x² + 6x + 5 =
x² - 5x -14 = 
x² + 6x -27 =
x² - 20x + 36 =
A

x² + 6x + 5 = (x+5)(x+1)

x² - 5x -14 = (x+2)(x-7)

x² + 6x -27 = (x+9)(x-3)

x² - 20x + 36 = (x-2)(x-18)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q 
Factor :
6x^3  -  150x =
2x² - 22x +48 =
7x^4 - 56x^3 - 63x² =
-3x^9 + 48xy^4 =
A

6x^3 - 150x = 6x ( x² -25 ) = 6x (x-5) (x+5)

2x² - 22x +48 = 2 (x² - 11x +24) = 2 ( x - 8 ) ( x + 3 )

7x^4 - 56x^3 - 63x² = 7x² (x² - 8x - 9) = 7x² (x-9)(x+1)

-3x^9 + 48xy^4 = 3x (16y^4 - x^8) = 3x (4y² + x^4)(4y² - x^4) = 3x (4y² + x^4) (2y + x²) (2y - x²)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Find the prime factorization of 1599, 2491, 9975

A

notice that
1599 = 1600 - 1 = 40² - 1² = (40 + 1)(40 - 1) = (41)(39) =(41)(313)
413*13

2941 = 2500 - 9 = 50²- 3² = ( 50 -3)(50+3) = 4753
4753

9975 = 10 000 - 25 = 100² - 5² = (105)(95) = (521)(519) = (537)(519)
5²3719

0.9991 = 1 - 0.0009 = 1² - 0.03² = (1+0.03)(1-0.03) = (1.03)(0.97)

EXPRESS THE NB AS THE DIFFERENCE OF TWO SQARES

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Simplify :

  1. 999951 / 0.993
    (0. 999856/0.988) - 1
A

0.999951 / 0.993 = (1 - 0.000049) / (1 - 0.007)
= (1² - 0.007²) / (1 - 0.007)
= (1 + 0.007)(1 - 0.007) / (1 - 0.007)
= 1.007

(0.999856/0.988) - 1 = [(1 - 0.000144)/(1 - 0.012)] - 1
= [(1 - 0.012)(1 + 0.012) / (1 - 0.12)] - 1
= 1.012 - 1
= 0.012

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Simplify :
( x² + 4x - 21) / ( x² - 6x + 9)
( 2x^4 - 8x² ) / ( x² - 5x - 14)
( y² + 2x - 8 ) / ( x - 4 )

A

( x² + 4x - 21) / ( x² - 6x + 9) = [(x+7)(x-3)] / (x-3)²
= (x+7) / (x-3)

( 2x^4 - 8x² ) / ( x² - 5x - 14) = [2x²(x² - 4)] / [(x+2)(x-7)]
= [2x² (x+2)(x-2)] / [(x+2)(x-7)]
= [2x²(x-2)] / (x-7)

( y² + 2x - 8 ) / ( x - 4 ) = [( y² / (x-4)] + [2(x-4)) / (x-4)]
= [y²/(x-4)] + 2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

If (x² + 12x - 45) / ( x + 15) = 22, then x = ?

A

(x² + 12x - 45) / ( x + 15) = 22
(x+15)(x-3) / (x+15) = 22
x - 3 = 22
x = 25

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

If ( 4P + 3Q - 4R) / ( P - R ) = 19, then Q / (P - R) = ?

A 
( 4P +  3Q - 4R) / ( P - R ) = 19
[4( P - R) + 3Q] / ( P - R ) = 19
4 + [3Q / (P-R)] = 19
3Q/(P-R) = 15
Q / (P - R) = 5
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Simplify :

[ (x²/7) - (10x/7) + 3] / [ (x/6) - (1/2) ] = ?

A

multiply by the LCM of 6, 7 and 2 which is 42
42 [ (x²/7) - (10x/7) + 3] / 42 [ (x/6) - (1/2) ]
6(x² - 10x + 21) / 7(x - 3)
6(x-3)(x-7) / 7(x-3)
6(x-7) / 7
(6/7) * (x-7)

17
Q

Solve :
x² + 13x - 40 = 8
3x² - 9x - 70 = 14
4x² - 4x - 4 = 14 + 8x - 2x²

A

x² + 13x - 40 = 8
x = -16 OR x = 3

3x² - 9x - 70 = 14
x = -4 OR x = 7

4x² - 4x - 4 = 14 + 8x - 2x²
x = -1 OR x = 3

18
Q

Solve the system of equations :
A/ w - 2x + 3y = 13
B/ 2w + x - 4y = -14
C/ 3w - x + 2y = 8

A

pick two equations and eliminate one variable (substitution or elimination)
B + C = 5w - 2y = -6
5w = 2y - 6

pick another pair of equations and eliminate the same variable
A + 2B = 5w - 5y = -15
5w = 5y - 15

use the two equations with two unknowns technique
5y - 15 = 2y - 6
y = 3

5w = 2y - 6
w = 0

w - 2x + 3y = 13
x = -2

19
Q

5x + 7 < 2x - 2
2/x > 1/3
-4 < 5 - 3x < 17

A

5x + 7 < 2x - 2
3x + 7 < -2
3x < -9
x < -3

2/x > 1/3
notice that x must unequal 0 and x > 0 because all nb are positive, we can cross multiply
6 > x
0 < x < 6

-4 < 5 - 3x < 17
-9 < -3x < 12
3 > x > -4

+ and - OK
x and / by positive OK
x and / by negative : reverse the direction of the inequality

20
Q

What operations can we perform
if a < b and b < c ?
if a < b and c < d ?

A

if a < b and b < c :
we can combine inequalities in the same direction
a < b < c

if a < b and c < d :
we can add inequalities in the same direction
a + c < b + d
we can subtract inequalities in opposite directions
b - c < a - d
(no rule for multiplication or division)

21
Q

Express |x - 7| < 3 as an ordinary inequality

A

we start at 7 on the nb line, and can go as far as a distance of 3 from that starting point
7 - 3 = 4
7 + 3 = 10
those are the endpoints, everything between them is allowed
4 < x < 10

22
Q

Express the region -3 < x < 11 as an absolute value inequality

A

Find the middle of the region by averaging the endpoints
middle = 4
notice that x can be as far as 7 above 4 or 7 below 4
|x-4| < 7

23
Q

Solve for x :

(2x - 1)² + 5(2x - 1) = 24

A

(2x - 1)² + 5(2x-1) - 24 = 0

Let u = 2x - 1
u² + 5u - 24 = 0
(u + 8)(u - 3) = 0
u + 8 = 0 OR u - 3 = 0
u = -8 OR u = 3

Use these values of u to solve for x
-8 = 2x - 1 OR 3 = 2x -1
x = -7/2 OR x = 2

24
Q

Solve for x :

(x²+1)² - 15(x²+1) + 50 = 0

A 
Let u = x²+1
u² - 15u + 50 = 0
(u - 5)(u - 10) = 0
u - 5 = 0 OR u - 10 = 0
u = 5 OR u = 10

x² + 1 = 5 OR x² + 1 = 10
x² = 4 OR x² = 9
x = +/- 2 OR x = +/- 3

25
Q

Solve for k:

3 / [ 1 - 8/(7 + k)] = 15

A

Let A = 1 - 8/(7 + k)
Then 3/A = 15 so 3 = 15 A and A = 1/5

Let B = 8 / (7+k)
Then A = 1 - B so 1/5 = 1 - B and B = 1 - 1/5 = 4/5

Then 4/5 = 8 / (7+k)
so 1/5 = 2 / (7+k)
7 + k = 10
k = 3

GMAT (11 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions