Algebra

Question 1

x = 3y - 2

10 - 2x = 8 - 3y

Solve for x and y using Substitution

Answer 1

We substitute the right side of the 1st equation in for “x” in the 2nd equation.

Make sure to use Parenthesis when doing substitution!

10 - 2(3y - 2) = 8 - 3y

10 - 6y + 4 = 8 - 3y

14 = 8 + 3y

6 = 3y

y = 2

Plug y = 2 back into 1st equation, to solve for x: x = 3(2) - 2 = 4

Question 2

What is a Quadratic Equation?

What is the General Form of a Quadratic Equation?

How would we put x² - 3x = 2 into General Form, so we can factor it?

Answer 2

A Quadratic equation means that the variable is taken to the 2nd power.

Quadratics often have 2 solutions! Watch out– don’t assume only 1 solution!

The General Form is ax² + bx + c = 0, where a, b, and c are constants

Subtract 2 from both sides:

x² - 3x - 2 = 0

Question 3

Factor x² - 3x - 28 = 0

Answer 3

Question 4

Solve by factoring:

x² - x - 12 = 0

Answer 4

(x - 4) (x + 3) = 0

x = 4 or x = -3

To check, we can use FOIL:

F = First = x*x = x²

O = Outer = x*3 = 3x

I = Inner = -4x

Outer + Inner combines to = 3x - 4x = -x

L= Last = -4*3 = -12

Question 5

Solve by factoring:

x² + 10x = 0

Answer 5

Factor out an “x” from each term:

x(x + 10) = 0

x = 0 or (x + 10) = 0

x = 0 or -10

Question 6

x² = 25

x = ?

Answer 6

x = 5 OR -5

Remember to look for 2 solutions on Quadratics!

Question 7

x² = 2x

Answer 7

This is a “Disguised Quadratic”. It looks different from the General Form. However, it should be manipulated, factored, and solved as a Quadratic (which often gives 2 solutions).

x² - 2x = 0

x(x - 2) = 0

x = 2 or 0

Common Error (we only get one solution. We miss the x=0 solution):

Divide both sides by x: x = 2

This is similar to the following error: x² = 16, square root both sides, x = 4

OR, x = -4

Question 8

Answer 8

Question 9

Factor x² - y²

This is very common on GMAT. What do we call it?

Answer 9

Difference between Squares:

x² - y² = (x + y) (x - y)

Question 10

Factor:

4x² - 25

Answer 10

Difference between Squares:

(2x + 5) (2x - 5)

Question 11

Factor:

16x⁴ - 9y²

Answer 11

Difference between Squares:

(4x² + 3y) (4x² - 3y)

Question 12

Factor x² + 2xy + y²

Answer 12

1: “Difference Between Squares”: x² - y² = (x + y) (x - y)

(x + y) (x + y) = (x + y)²

This results in only 1 solution, since it’s a perfect square.

These 3 Special Products are very common:

Question 13

Factor x² - 2xy + y²

Answer 13

1: “Difference Between Squares”: x² - y² = (x + y) (x - y)

(x - y) (x - y) = (x - y)²

This results in only 1 solution, since it’s a perfect square.

These 3 Special Products are very common:

Question 14

Solve by factoring:

x² + 9 = -6x

Answer 14

Add 6x to both sides: x² + 6x + 9 = 0

(x + 3) (x + 3) = 0

Can also be written as (x+3)² = 0

x = -3

Only 1 solution, because it’s a perfect square

Question 15

Factor: a²b² + ab = 2

Then, find b in terms of a

Answer 15

First, subtract 2 from both sides:

a²b² + ab - 2 = 0

(ab + 2) (ab - 1) = 0

To check, we can FOIL:

F: ab*ab = a²b²

OI: ab* -1 + ab*2 = ab

L: 2*-1 = -2

To find b in terms of a: (ab+2) = 0 OR (ab - 1) =0

(ab+2) = 0, ab = -2, so b = -2/a OR

(ab - 1) = 0, ab = 1, so b = 1/a

Question 16

What is |-10| ?

Answer 16

10

It turns a negative value into a positive.

| is the absolute value symbol.

Question 17

|x| = 4

What is x?

Answer 17

x = 4 or -4

Question 18

Answer 18

Question 19

Data Sufficiency:

What is the value of x + 2y ?

(1) x + y = 2
(2) 6x + 12y - 3 = 15

Answer 19

(1) is Insufficient.
(2) is Sufficient.

We don’t always have to know both variables individually, when the question is asking for a combined value.

Watch out for the “C Trap”: when we think we need both statements, but one statement is sufficient.

We can manipulate (2), so that the left side is the same as the question:

Add 3 on both sides: 6x + 12y = 18

Divide by 6 on both sides: x + 2y = 3

Question 20

Simplify, by splitting fraction into 2 terms:

Answer 20

Question 21

Answer 21

Question 22

What is 4 squared?

Answer 22

4 squared = 4² = 4 * 4 = 16

4 is the base. 2 is the power.

For a power of 2, we use the word “squared”.

(This is because a square with 4 units on each side has an area of 4²)

Question 23

What is 2 cubed?

Answer 23

2 cubed = 2 * 2 * 2 = 2³ = 8

For a power of 3, we use the word “cubed”.

(This is because a cube with 2 units on each side has an area of 2³)

Question 28

Answer 28

Question 29

Answer 29

Question 30

What “Perfect Square” less than 100 has 5 as a prime factor?

Answer 30

25 = 5²

a Perfect Square is the square of an integer

For a perfect square, the prime factors have even exponents:

Examples: 36 = 2² * 3²

100 = 2² * 5²

Question 31

What “Perfect Cube” less than 100 has 3 as a prime factor?

Answer 31

3³ = 27

a Perfect Cube is the cube of an integer

For a perfect cube, the prime factors have powers that are a multiple of 3.

Example: 1728 = 2⁶ * 3³ = (2*2*3)³ = 12³

Question 32

2³ * 2⁴ = 2^?

Answer 32

Multiplying Like Bases:

2³ * 2⁴ = 2⁷

^{Add the powers:}

x^a * x^b = x^a+b

Question 33

Answer 33

Question 34

What is 0.3³, as a fraction?

Answer 34

Question 35

x^y-2 * x^3y+5 = x^?

Answer 35

x^y-2 * x^3y+5 = x^4y+3

^{Multiplying Like Bases: add the powers}

x^a * x^b = x^a+b

Question 38

(x²)³ = x^?

Answer 38

(x²)³ = x⁶

Power to a Power Rule:

(x^{<span>a</span>})^{<span>b</span>} = x^ab

Question 39

a) What is x^-1 ?
b) What is 2^-3 ?

Answer 39

A negative exponent is the Reciprocal of the positive exponent:

(1 divided by the positive exponent)

Question 40

a) Express using a negative exponent
b) Express using a positive exponent

Answer 40

Question 41

If x^-1 = 5, what is x?

Answer 41

Question 42

Answer 42

Square Root is the inverse of Squaring.

Question 43

Answer 43

Question 44

Answer 44

Question 45

Answer 45

Question 46

Answer 46

We can apply the square root separately, to x², and to y⁴

We get |x|*y²

We need an absolute value sign around the x, when we take a square root of a square, to account for the fact that x could be positive or negative.

For example, the square root of 3² is 3, and the square root of (-3)² is also 3.

The absolute value sign is not necessary around the y², because y² cannot be negative.

Question 47

Answer 47

Question 48

Answer 48

Question 49

0^a = ?

Answer 49

0

0 to any power is 0.

Question 50

1^a = ?

Answer 50

1

1 to any power is 1

Question 53

Is 0.4³ > 0.4² ?

Why or why not?

Answer 53

No.

A fraction between 0 and 1 gets SMALLER as the power increases.

0. 4³ = 64/1000 = .064
1. 4² = 16/100 = .16

Question 54

Is -(1/2)³ > -1/2 ?

Answer 54

Yes.

A positive fraction between 0 and 1, taken to greater power, gets closer to 0, and therefore smaller.

A negative fraction between -1 and 0, taken to an odd power,

is still negative and gets closer to 0, and therefore greater.

(-1/2)³ = -1/8

-1/8 > -1/2

Question 55

is a/b > (a/b)² ?

(1) a < b
(2) a and b are positive

Answer 55

1) Insufficient. If a = 1 and b = 2, 1/2 > 1/4 –> Yes

If a = -1 and b=2, -1/2 < 1/4 –> No

(Note that a positive fraction between 0 and 1 gets smaller when raised to a higher power, but also we have to account for the possibility of a negative number)

2) Insufficient. If a = 1 and b = 2, 1/2 > 1/4 –> Yes

If a = 2 and b = 1, 2 No

Together: Sufficient. (2) rules out the negative number possibility that made (1) Insufficient

Question 56

Positive or negative?

a. (-1)⁵ ?
b. (-2)³
c. -3² ?
d. -3*(-4)⁷ ?

Answer 56

a. Negative. A negative number taken to an odd power is negative.
b. Negative.
c. Negative. Without the parentheses, we have to take the exponent first, before the - sign.

So, 3² = 9, and -3² = -9

d. Positive. (-4)⁷ is negative, so -3*(-4)⁷ is positive