Module 3: Linear and Quadratic Equations Flashcards
Linear equation (definition)
- 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.
Solving Linear Equations with Two Variables: 2 strategies
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
When to use Substitution vs. Elimination method
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
Eliminating fractions from an equation
Multiply the entire equation by the least common denominator: the smallest number that all denominators will divide into
Using LCD to Simplify: A note
When multiplying the entire equation by X to simplify, DON’T distribute the X to values inside parentheses
Common Factor strategy
When all variables in an equation have a common factor, try FACTORING IT OUT (don’t just divide it out) to simplify
Zero Product Property + a catch
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
If I see that the product of two integers is 1…
I will know that either both are 1 or both are -1.
Quadratic Equations & Factoring
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.
Quadratic Equations & FOILing
From factored form, multiply First, Outside, Inside & Last terms
(x - 7) (x + 4)
x^2 - 3x - 28
Quadratic Identities
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
FOIL: (x + y)^2
(x + y)^2 = (x + y)(x + y) = x^2 + y^2 + 2xy
Factor: x^2 + y^2 + 2xy
x^2 + y^2 + 2xy = (x + y)(x + y) = (x + y)^2
FOIL: (x - y)^2
(x - y)^2 = (x - y) (x - y) = x^2 + y^2 - 2xy
Factor: x^2 + y^2 - 2xy
x^2 + y^2 - 2xy = (x - y) (x - y) = (x - y)^2
FOIL: (x + y) (x - y) (what is this called?)
(x + y) (x - y) = x^2 - y^2
The Difference of Two Squares
Factor: x^2 - y^2 (what is this called?)
x^2 - y^2 = (x + y) (x - y)
The Difference of Two Squares
Spotting the Difference of Squares
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)
Exponent Power of Power Rule
(x^a)^b = x^ab
Use to match exponents in an equation, and then simplify.
If x != y:
(x - y) / (y - x) =
-1
Common GMAT Data Sufficiency Equation Traps (6)
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.
When I am stuck on math for a problem solving question with an equation with multiple variables, I will
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.
Solving Cubic or Quartic Equations: 3 steps (Normal method)
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
Factoring by Grouping (3? steps)
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
When I am about to divide by a variable I will
Ask myself: “do I know that X is not equal to zero?” If not, then you CAN’T DO DIVISION