Module 3: Linear and Quadratic Equations

Question 1

Linear equation (definition)

Answer 1

• An equation with one or more variables
• Each variable raised only to the first power
• Variables are not multiplied together or divided by each other.

Question 2

Solving Linear Equations with Two Variables: 2 strategies

Answer 2

Substitution Method: Isolate one of two variables in either equation & insert that value into the other equation.

Elimination Method: Combine equations with addition or subtraction in order to eliminate all but one variable. May require further manipulation: multiplication or division of one or more sides to match coefficients, etc

Question 3

When to use Substitution vs. Elimination method

Answer 3

Substitution: If one of the equations can easily be manipulated to isolate a variable on one side of the equation

Elimination: If neither equation can be easily manipulated to solve one of the variables

Question 4

Eliminating fractions from an equation

Answer 4

Multiply the entire equation by the least common denominator: the smallest number that all denominators will divide into

Question 5

Using LCD to Simplify: A note

Answer 5

When multiplying the entire equation by X to simplify, DON’T distribute the X to values inside parentheses

Question 6

Common Factor strategy

Answer 6

When all variables in an equation have a common factor, try FACTORING IT OUT (don’t just divide it out) to simplify

Question 7

Zero Product Property + a catch

Answer 7

If product of two quantities equals 0, then at least one of the quantities must be 0.

CATCH: Don’t assume that the variable is not zero!
Example:

x(x + 100) = 0

X = 0
OR
X = - 100

Question 8

If I see that the product of two integers is 1…

Answer 8

I will know that either both are 1 or both are -1.

Question 9

Quadratic Equations & Factoring

Answer 9

An equation in which the highest power of an unknown quantity is 2.

Factor form: ax^2 + bx + c

Solution: (x + p) (x + q)
P & Q must MULTIPLY to yield C
P & Q must ADD to yield B

The final answer will be P and Q with their parity signs flipped.

Question 10

Quadratic Equations & FOILing

Answer 10

From factored form, multiply First, Outside, Inside & Last terms

(x - 7) (x + 4)

x^2 - 3x - 28

Question 11

Quadratic Identities

Answer 11

Statements that are true for all possible values of a variable X. There are three common Quadratic Identities that should be memorized for the GMAT:

1 - (x + y)^2 = (x + y)(x + y) = x^2 + y^2 + 2xy
2 - (x - y)^2 = (x - y) (x - y) = x^2 + y^2 - 2xy
3 - (x + y) (x - y) = x^2 - y^2

Question 12

FOIL: (x + y)^2

Answer 12

(x + y)^2 = (x + y)(x + y) = x^2 + y^2 + 2xy

Question 13

Factor: x^2 + y^2 + 2xy

Answer 13

x^2 + y^2 + 2xy = (x + y)(x + y) = (x + y)^2

Question 14

FOIL: (x - y)^2

Answer 14

(x - y)^2 = (x - y) (x - y) = x^2 + y^2 - 2xy

Question 15

Factor: x^2 + y^2 - 2xy

Answer 15

x^2 + y^2 - 2xy = (x - y) (x - y) = (x - y)^2

Question 16

FOIL: (x + y) (x - y) (what is this called?)

Answer 16

(x + y) (x - y) = x^2 - y^2

The Difference of Two Squares

Question 17

Factor: x^2 - y^2 (what is this called?)

Answer 17

x^2 - y^2 = (x + y) (x - y)

The Difference of Two Squares

Question 18

Spotting the Difference of Squares

Answer 18

Look for the square of a value minus the square of another value.

x^2 - y^2

They might not both be in variable form…

100 - 4x^2 (squares of 10 and 2x)
x^2 - 1 (squares of x and 1)
1/36 x^2 - 25 (squares of 1/6x and 5)
x^2y^2 - 16 (squares of xy and 4)
3^30 - 2^30 (squares of 3^15 and 2^15)

Question 19

Exponent Power of Power Rule

Answer 19

(x^a)^b = x^ab

Use to match exponents in an equation, and then simplify.

Question 20

If x != y:

(x - y) / (y - x) =

Answer 20

-1

Question 21

Common GMAT Data Sufficiency Equation Traps (6)

Answer 21

1) Two equations look different, but actually provide the same information (common in DS)
2) One equation can be sufficient to solve 2 different variables - other limitations in the problem, such as integer constraints, may restrict the possible values enough that an answer can be determined
3) One equation can be substituted into another to solve - common for “solve the combo” DS questions
4) There is a hidden quadratic equation, AND/OR the quadratic has only one solution.
5) An equation has three or more solutions. (Normally involves a cubic or quartic term)
6) Variables - as well as more complex variable EXPRESSIONS - can be equal to zero! If you don’t know for sure that it isn’t, then you can’t divide by it.

Question 22

When I am stuck on math for a problem solving question with an equation with multiple variables, I will

Answer 22

try solving the equation for one of the variables - that might reveal something important. There MUST be a solution - there is probably a number properties trick that you aren’t thinking of.

Question 23

Solving Cubic or Quartic Equations: 3 steps (Normal method)

Answer 23

1) Move all terms to one side, set equal to zero
2) Factor out variables from that side
3) Solve for the difference of squares.

x^2 - y^2 = (x - y)(x + y)

Solutions = +y, -y

4) Add “0” as the third possible solution

Question 24

Factoring by Grouping (3? steps)

Answer 24

Commonly used when one side of an equation is zero and there are four total terms.

1) Factor variables in groups that share a common factor:

x^3 - x^2 - 81x + 81 = 0
x^2(x - 1) - 81(x - 1) = 0

2) Now you can factor out the newly visible (x - 1):
(x - 1) (x^2 - 81) = 0

3) Difference of squares:
(x - 1) (x - 9) (x + 9) = 0

x = 1, 9, -9

Question 25

When I am about to divide by a variable I will

Answer 25

Ask myself: “do I know that X is not equal to zero?” If not, then you CAN’T DO DIVISION