ALEKS Math flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Slope formula

The slope of a line through points (x1,y1) and (x2,y2) is (y2 - y1)/(x2 - x1). It measures vertical change per unit horizontal change.

Quadratic formula

For ax^2 + bx + c = 0 with a not zero, x = (-b ± sqrt(b^2 - 4ac))/(2a). It gives all real or complex roots of the equation.

Discriminant

The value b^2 - 4ac in a quadratic. Positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Product rule for exponents

When multiplying powers with the same base, add the exponents: a^m * a^n = a^(m+n). The base stays unchanged.

Negative exponent rule

A base raised to a negative exponent equals the reciprocal raised to the positive exponent: a^(-n) = 1/a^n, for a not zero.

Logarithm product rule

The log of a product equals the sum of the logs: log(ab) = log a + log b. This converts multiplication into addition.

Percent change formula

Percent change equals (new value - old value) divided by old value, times 100. A positive result is an increase.

Distance formula

The distance between two points is sqrt((x2-x1)^2 + (y2-y1)^2). It follows from the Pythagorean theorem on coordinate differences.

Midpoint formula

The midpoint of a segment with endpoints (x1,y1) and (x2,y2) is ((x1+x2)/2, (y1+y2)/2), the average of the coordinates.

Pythagorean theorem

In a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse. It holds only for right triangles.

Circle area and circumference

For radius r, area = pi*r^2 and circumference = 2*pi*r. Diameter equals 2r, so circumference also equals pi*d.

Order of operations

Evaluate parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Point-slope form

A line through (x1,y1) with slope m is y - y1 = m(x - x1). Rearranging gives slope-intercept form y = mx + b.

Vertical line slope

A vertical line has undefined slope because the run (x2 - x1) is zero, making division by zero. A horizontal line has slope zero.

Function definition

A relation is a function if each input has exactly one output. The vertical line test checks this on a graph.

Absolute value rule

The absolute value |x| is the distance from x to zero, always nonnegative. Equations like |x| = k have solutions x = k and x = -k.

Mole conversion

Moles equal mass in grams divided by molar mass in g/mol. One mole contains 6.022 x 10^23 particles (Avogadro's number).

Molarity

Molarity M equals moles of solute divided by liters of solution. It expresses solution concentration and is used in dilution calculations.

Dilution equation

M1*V1 = M2*V2 relates concentration and volume before and after dilution. Moles of solute stay constant when solvent is added.

Ideal gas law

PV = nRT relates pressure, volume, moles, and temperature. R = 0.08206 L*atm/(mol*K) and temperature must be in kelvin.

Kelvin conversion

Kelvin equals degrees Celsius plus 273.15. Gas-law calculations require absolute temperature because volume scales from absolute zero.

pH definition

pH = -log[H+], where [H+] is hydrogen ion concentration in mol/L. A pH below 7 is acidic at 25 °C.

pH and pOH relation

At 25 °C, pH + pOH = 14 and [H+][OH-] = 1.0 x 10^-14. pOH equals -log[OH-].

Percent yield

Percent yield equals actual yield divided by theoretical yield, times 100. Theoretical yield comes from stoichiometry of the limiting reactant.

Limiting reactant

The reactant that runs out first and limits product formation. It is found by comparing mole ratios of available reactants to the balanced equation.

Balanced equation rule

A balanced equation has equal numbers of each type of atom on both sides. Coefficients give mole ratios used in stoichiometry.

Significant figures rules

Nonzero digits and zeros between them count. Leading zeros never count. Trailing zeros count only when a decimal point is present.

Density formula

Density equals mass divided by volume, commonly in g/mL or g/cm^3. It can be rearranged to find mass or volume.

Molar volume at STP

At standard temperature and pressure (0 °C, 1 atm), one mole of ideal gas occupies 22.4 liters.

Oxidation and reduction

Oxidation is loss of electrons and reduction is gain of electrons. The oxidizing agent accepts electrons and is itself reduced.

Avogadro's number

6.022 x 10^23 is the number of particles in one mole of a substance. It converts between particle count and moles.

Molar mass

The mass in grams of one mole of a substance, equal to the sum of atomic masses in its formula. Units are g/mol.