Algebra: Quadratic Equations and Formulas

Linear growth (or decay)

y = mx + b m = constant rate at which the quantity grows y-intercept b = value of the quantity at time 0 (assuming time is x) x = time

Identify patterns in sequence problems.

"If S(n) = 3ⁿ, what is the units digit of S(65)?" Clearly you cannot multiply out 3⁶⁵. Look for a pattern in the units digit of the powers of 3!

For linear sequences, in which the same number is added to any term to yield the next term, consider an alternative/aggregate method...

"If each number in a sequence is 3 more than the previous number, and the 6th number is 32, what is the 100th number?" From the 6th to the 100th term, there are 94 "jumps" of 3. 32 + 94*3 = 314

Population problems

"The population of a bacterium triples every 10 minutes..." MAKE A TABLE with a few rows, labeling one row NOW.

x² - y² = (special product #1)

(x + y)(x - y)

x² + 2xy + y² = (special product #2)

(x + y)²

x² - 2xy + y² = (special product #3)

(x - y)²

Four major types of formula problems:

1. Plug-in formulas 2. Functions 3. Strange symbol formulas 4. Sequence formulas (can be recursive)

See Data Sufficiency Cheat Sheet for combo problems.

Hint to save time: as a general rule, if you are given just one linear equation with only basic math operations (+, -, *, ÷), you cannot alter the initial relationship between the two variables

Be careful with DISGUISED quadratics, e.g., 3w² = 6w

If you solve this equation without factoring it like a quadratic, you will miss one of its solutions!! 3w² - 6w = 0 3w(w - 6) = 0 Therefore, w = {0, 2} Other disguised examples are 36/b = b-5 and x³ + 2x² -3x = 0 (<===== this expression has three solutions)

Follow the formula's instructions ________, especial for a formula or special symbol that acts on decimals.

PRECISELY

Common function types

Population problems Proportionality (direct and inverse) Linear growth Symmetry

Strategy when factoring a quadratic equation

Set the equation equal to 0. Assuming that a = 1 (if not, divide through), you need to find two integers whose product is c and whose sum is b

Symmetry

These are questions about whether two seemingly different inputs to a function always yield the same output (e.g., "for which of the following functions does f(x) = f(1/x)?"). Rather than figure out the answer algebraically, make a chart, testing A through E for f(x) and f(1/x), checking for symmetry.

Standard form of a quadratic equation

ax² + bx + c = 0

If you encounter a quadratic equation or expression, try ________ it. On the other hand, if you encounter the product of factors such as (x + 7)(x + 3), you may need to ________ it.

factoring, FOIL

One-Solution Quadratics

happens when both roots are the same (x + 4)² = 0 (x - 3)² = 0 BL: when you see a quadratic, look for two solutions, but know that sometimes there's just one

Domain (of a function)

indicates possible inputs

Range (of a function)

indicates possible outputs

When taking the square root of a quadratic, consider both _______________.

the positive and the negative square root (z + 3)² = 25 z = {2, -8}

"Less than" on a number line means "___________".

to the left of I.e., a "smaller" negative number is further away from 0 than a "bigger" negative number.

Like other even exponent equations, quadratic equations generally have ________ solutions.

two

Indirect proportionality

y = k/x (two quantities change by reciprocal factors) y₁*x₁ = y₂*x₂ (can use smart numbers)

Direct proportionality

y = kx (two quantities change by the same factor in the same direction) y₁/x₁ = y₂/x₂