Algebra Basic Flashcards
Slope formula
m = (y₂ - y₁) / (x₂ - x₁)
Quadratic formula
x = (-b ± √(b² - 4ac)) / 2a
Distance formula
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Midpoint formula
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Pythagorean theorem
a² + b² = c²
Slope-intercept form
y = mx + b
Standard form of a linear equation
Ax + By = C
Point-slope form
y - y₁ = m(x - x₁)
Difference of squares formula
a² - b² = (a - b)(a + b)
Factored form of a quadratic equation
a(x - r₁)(x - r₂)
FOIL method
(a + b)(c + d) = ac + ad + bc + bd
Vertex form of a quadratic equation
y = a(x - h)² + k
Discriminant formula
b² - 4ac
Equation of a circle
(x - h)² + (y - k)² = r²
Exponential growth formula
A = A₀ e^(rt)
Logarithm rule: logₐ(xy)
logₐ x + logₐ y
Logarithm rule: logₐ(x/y)
logₐ x - logₐ y
Logarithm rule: logₐ(xᵇ)
b logₐ x
Sum of an arithmetic sequence
Sₙ = (n/2) (a + l)
General form of a quadratic equation
ax² + bx + c = 0
Sum of the roots of a quadratic equation
-b/a
Product of the roots of a quadratic equation
c/a
Equation of a line given two points (x₁, y₁) and (x₂, y₂)
y - y₁ = (y₂ - y₁) / (x₂ - x₁) * (x - x₁)
Exponential decay formula
A = A₀ e^(-rt)
Compound interest formula
A = P(1 + r/n)^(nt)
Simple interest formula
I = PRT
Formula for the nth term of an arithmetic sequence
aₙ = a₁ + (n - 1)d
Formula for the nth term of a geometric sequence
aₙ = a₁ * r^(n-1)
Sum of a finite geometric series
Sₙ = a₁(1 - rⁿ) / (1 - r)
Sum of an infinite geometric series (|r| < 1)
S = a₁ / (1 - r)
Logarithmic identity: logₐ(1)
0
Logarithmic identity: logₐ(a)
1
Change of base formula
logₐ b = log_c b / log_c a
Power rule for exponents
(a^m)^n = a^(m*n)
Product rule for exponents
a^m * a^n = a^(m+n)
Quotient rule for exponents
a^m / a^n = a^(m-n)
Zero exponent rule
a⁰ = 1 (for a ≠ 0)
Negative exponent rule
a^(-n) = 1 / a^n
Radical to exponent conversion
√a = a^(1/2)
Variable definition
A symbol (often x) that represents an unknown or changeable value
Constant definition
A fixed value that does not change
Coefficient definition
A numerical factor multiplying a variable (e.g., 5 in 5x)
Term definition
A single mathematical expression (e.g., 3x, -7, 2y²)
Like terms definition
Terms with the same variables to the same powers (e.g., 2x and 5x)
Expression vs. Equation
Expression: A combination of terms (3x + 5); Equation: Two expressions set equal (3x + 5 = 14)
Distributive property formula
a(b + c) = ab + ac
Commutative property of addition
a + b = b + a
Associative property of multiplication
(ab)c = a(bc)
Additive inverse definition
The opposite of a number (a + (-a) = 0)
Combining like terms rule
Add/subtract coefficients of terms that have identical variable parts
Simplify 3x + 7 + 2x - 4
5x + 3
Factor out the greatest common factor (GCF) in 8x + 12
4(2x + 3)
Expand (x + 2)(x + 3)
x² + 5x + 6
Simplify 2(x + 4) - 3(x - 1)
-x + 11
Explain FOIL method
Multiply First terms, Outer terms, Inner terms, Last terms in two binomials
Rewrite 12x - 6y using factoring
6(2x - y)
Expression for perimeter of a rectangle (length x and width y)
P = 2x + 2y
Expression for area of a triangle (base b and height h)
A = (1/2)bh
Definition of polynomial
An expression of finite terms with nonnegative integer exponents (e.g., x² - 4x + 7)
Linear equation standard form
Ax + By = C
Slope-intercept form
y = mx + b
Slope formula between points (x₁, y₁) and (x₂, y₂)
m = (y₂ - y₁) / (x₂ - x₁)
Y-intercept definition
The point where a line crosses the y-axis (x=0)
Solve for x: 3x - 9 = 0
x = 3
Solve for x: x/2 + 2 = 5
x = 6
Inequality solution example: Solve x + 3 > 7
x > 4
Graph of y = x + 2
A straight line with slope 1 and y-intercept (0, 2)
Solve for y: -2y + 4 = 10
y = -3
Rewrite the inequality x < -2 in interval notation
(-∞, -2)
Definition of a system of equations
Two or more equations with the same variables
Solution to a system of equations
A set of variable values satisfying all equations simultaneously
Substitution method
Solve one equation for a variable, then substitute into the other equation
Elimination method
Add or subtract equations to eliminate a variable
Solve by substitution: x + y = 4 and y = 2x
x = 4/3; y = 8/3
Solve by elimination: 2x + y = 7, x - y = 1
x = 8/3; y = 5/3
Graphical method definition
Plot each line; the intersection point gives the solution
No solution to a system
Lines are parallel with different intercepts (inconsistent system)
Infinite solutions to a system
Lines are coincident (the same line)
Solve system: x + y = 2, x + y = 5
No solution (parallel lines)
Product rule of exponents
a^m · a^n = a^(m+n)
Quotient rule of exponents
a^m / a^n = a^(m-n) (a≠0)
Power rule of exponents
(a^m)^n = a^(mn)
Zero exponent rule
a^0 = 1 (a≠0)
Negative exponent rule
a^(-n) = 1 / a^n
Simplify: x² · x³
x⁵
Simplify: (x³)²
x⁶
Square root of x in radical form
√x = x^(1/2)
Cube root of x in radical form
∛x = x^(1/3)
Rational exponent to radical conversion
x^(m/n) = n√(x^m)
Sum of cubes factorization
a³ + b³ = (a + b)(a² - ab + b²)
Difference of cubes factorization
a³ - b³ = (a - b)(a² + ab + b²)
Difference of squares factorization
a² - b² = (a - b)(a + b)
Trinomial quadratic factorization example
x² + 5x + 6 = (x + 2)(x + 3)
Factor 3x² + 6x
3x(x + 2)
Factor 2x² - 8x + 6
2(x - 3)(x - 1)
Expand (x - 4)(x + 2)
x² - 2x - 8
Factor x² - 16
(x - 4)(x + 4)
Factor x² + x - 6
(x + 3)(x - 2)
Solve x² - 9 = 0 by factoring
(x - 3)(x + 3) = 0 → x=3 or x=-3
What is a variable in algebra?
A letter or symbol that represents an unknown or changeable value.
What is a constant in algebra?
A fixed value that does not change.
What is a coefficient?
A numerical factor multiplied by a variable (e.g., 7 in 7x).
Define a ‘term’ in an expression.
A single mathematical entity (e.g., 5, 3x, or -2y²).
What is an algebraic expression?
A combination of terms (variables, constants, operations) but no equals sign.
What distinguishes an equation from an expression?
An equation has an equals sign, an expression does not.
What is a polynomial?
An expression with one or more terms, each having nonnegative integer exponents on variables.
Define a monomial.
A polynomial with exactly one term (e.g., 6x²).
Define a binomial.
A polynomial with two distinct terms (e.g., x + 3).
Define a trinomial.
A polynomial with three distinct terms (e.g., x² + 5x + 6).
What is the degree of a polynomial?
The highest sum of exponents on any term (e.g., degree of 4x³y is 4).
Write the standard form of a polynomial in x.
List terms in descending powers of x (e.g., 3x³ - 2x² + x + 1).
Define a zero (root) of a polynomial.
A value of x that makes the polynomial equal to zero.
What is a linear equation in one variable?
An equation of the form ax + b = 0, where a ≠ 0.
What is the slope-intercept form of a line?
y = mx + b, where m is slope and b is y-intercept.
State the slope formula between (x₁, y₁) and (x₂, y₂).
m = (y₂ - y₁) / (x₂ - x₁).
What is the standard form of a line?
Ax + By = C.
What is the point-slope form of a line?
y - y₁ = m(x - x₁).
Define ‘slope’ in coordinate geometry.
The ratio of vertical change to horizontal change: rise/run.
What is a y-intercept?
The point where the graph crosses the y-axis (x=0).
What is an x-intercept?
The point where the graph crosses the x-axis (y=0).
Define ‘domain’ of a function f(x).
All possible x-values for which f(x) is defined.
Define ‘range’ of a function f(x).
All possible output values f(x) can take.
What is a function in algebra?
A relation where each input has exactly one output.
Explain the vertical line test in graphs.
If any vertical line intersects a graph more than once, it’s not a function.
What is meant by composite function (f ∘ g)(x)?
It means f(g(x)), applying g first, then f.
Define the process of ‘factoring’ in algebra.
Rewriting an expression as a product of simpler expressions.
What is the Greatest Common Factor (GCF)?
The largest expression that divides all terms of a polynomial.
Write the difference of squares formula.
a² - b² = (a - b)(a + b).
Write the sum of squares factorization.
a² + b² is not factorable over the reals (no simple factorization).
Write the sum of cubes formula.
a³ + b³ = (a + b)(a² - ab + b²).
Write the difference of cubes formula.
a³ - b³ = (a - b)(a² + ab + b²).
How do you factor a quadratic expression x² + bx + c?
Find two numbers that multiply to c and add to b, then use them to split bx.
Give an example of factoring a trinomial: x² + 5x + 6.
x² + 5x + 6 = (x + 2)(x + 3).
Define a prime polynomial (over the reals).
A polynomial that cannot be factored further (other than trivial factors).
What does irreducible polynomial mean?
Same as prime polynomial: it cannot be factored over a given number system.
Define a ‘root’ or solution to an equation.
A value that satisfies the equation, making it true.
What does ‘zero of a polynomial’ mean?
A value of x for which the polynomial equals 0 (same as root).
Define an ‘extraneous solution.’
A solution that emerges from algebraic manipulation but doesn’t satisfy the original equation.
What is a system of equations?
Two or more equations with the same variables.
Define a ‘consistent system’ of linear equations.
A system that has at least one solution.
Define an ‘inconsistent system’ of linear equations.
A system that has no solutions.
Define a ‘dependent system’ of linear equations.
A system with infinitely many solutions; equations represent the same line.
What does f(x) mean in algebra?
Function notation: f is the function’s name, x is the input variable.
What is composition of functions, like f(g(x))?
Applying g first, then plugging the result into f.
State the quadratic formula for ax² + bx + c = 0.
x = [-b ± √(b² - 4ac)] / (2a).
What is the discriminant in a quadratic equation?
b² - 4ac.
What is the sum of the roots of ax² + bx + c = 0?
−b/a.
What is the product of the roots of ax² + bx + c = 0?
c/a.
Factoring x² - 16 uses which formula?
Difference of squares: x² - 16 = (x - 4)(x + 4).
What is the FOIL method?
First, Outer, Inner, Last multiplication for binomials.
What is the Binomial Theorem for (a + b)ⁿ?
Σ from k=0 to n of [nCk · a^(n-k) · b^k].
Define a rational expression.
A fraction with polynomials in numerator and denominator.
Why must we check domain restrictions for rational expressions?
Because denominator cannot be zero.
How do you simplify a rational expression?
Factor numerator and denominator, then cancel common factors.
How to add two rational expressions with denominators D1 and D2?
Use the common denominator D1·D2 (or LCM if simpler), then add numerators.
How to multiply two rational expressions?
Multiply numerators, multiply denominators, then simplify.
How to divide two rational expressions?
Multiply by the reciprocal of the second expression.
Define a radical expression.
An expression involving a root symbol (√, ∛, etc.).
What is the ‘radicand’ in √x?
The value under the radical sign (x).
Define an nth root, e.g., √[n]{x}.
A number that when raised to the nth power gives x.
What does a rational exponent x^(m/n) mean?
The nth root of x^m, or (x^m)^(1/n).
State the property √a × √b = √(ab).
Product of square roots = square root of the product.
State the property √(a/b) = √a / √b (b≠0).
Quotient under a radical = quotient of radicals.
What is the absolute value of a number?
Its distance from zero on the real number line.
Define an inequality in algebra.
A statement showing <, >, ≤, or ≥ between expressions.
Solve x + 3 > 5 in simple form.
x > 2.
Write x > 2 in interval notation.
(2, ∞).
What is a compound inequality?
Two inequalities joined by ‘and’ or ‘or’ (e.g., 1 < x < 3).
Define ‘domain’ of an inequality solution.
The set of all real numbers that satisfy that inequality.
State the reciprocal property: x × (1/x).
It equals 1 for x ≠ 0.
What does a negative exponent mean? x^(-n).
1/(x^n), for x ≠ 0.
What does (ab)^n mean?
a^n × b^n.
How do you combine like terms in an expression?
Add/subtract coefficients of terms with identical variable parts.
What are ‘like terms’?
Terms with the same variable(s) raised to the same powers.
State the commutative property of addition.
a + b = b + a.
State the commutative property of multiplication.
ab = ba.
State the associative property of addition.
(a + b) + c = a + (b + c).
State the associative property of multiplication.
(ab)c = a(bc).
State the identity property of addition.
a + 0 = a.
State the identity property of multiplication.
a × 1 = a.
What is the additive inverse property?
a + (-a) = 0.
What is the multiplicative inverse property?
a × (1/a) = 1, for a ≠ 0.
Interpret a^m / a^n (a≠0).
a^(m-n).
Interpret (a^m)^n.
a^(mn).
When do we use polynomial long division?
To divide one polynomial by another of equal or lower degree.
When do we use synthetic division?
To divide a polynomial by a binomial of the form (x - c).
What is the direct variation formula?
y = kx, for some constant k.
What is the inverse variation formula?
y = k/x, for some constant k (x≠0).
What is a joint variation formula?
z = kxy, for some constant k.
Difference between ‘expression’ and ‘equation’ in usage?
Expression is a phrase of math terms; equation is a statement with an equals sign.
Why use slope-intercept form y=mx+b?
It lets us easily read slope (m) and y-intercept (b).
What is the purpose of standard form Ax + By = C?
Often used for integer coefficients and analyzing intercepts quickly.
How do we use the slope formula?
To find the rate of change between two points on a line.
Difference between a function and a relation?
A function has each input mapped to exactly one output; a relation can have multiple.
Outline the steps in solving a 2-step linear equation, e.g., 2x + 3 = 9.
Subtract 3: 2x=6, then divide by 2: x=3.
What is the substitution method for solving a 2-variable system?
Solve one equation for one variable, then substitute into the other.
What is the elimination method for solving a 2-variable system?
Add or subtract equations to eliminate one variable, solve for the other.
How do you solve a linear system graphically?
Plot both lines; intersection point is the solution (if it exists).
How is factoring used to solve x² + bx + c = 0?
Factor into (x - r₁)(x - r₂)=0, so x=r₁ or x=r₂.
What is the general form of a linear equation in slope-intercept form?
y = mx + b
Write the formula for finding the x-intercept of a line y = mx + b.
Set y = 0, so x = -b/m
Write the formula for finding the y-intercept of a line Ax + By = C.
Set x = 0, then y = C/B
State the formula for the slope of a line in standard form Ax + By = C (B ≠ 0).
Slope m = -A/B
Give the formula for the slope of a line perpendicular to one with slope m.
Slope = -1/m
What is the formula for the line parallel to y = mx + b through (x₀, y₀)?
y - y₀ = m(x - x₀)
State the formula to convert a decimal r to a percentage.
Percentage = r × 100%
Give the point-slope form of a line through (x₁, y₁) with slope m.
y - y₁ = m(x - x₁)
Formula for an arithmetic sequence’s nth term?
aₙ = a₁ + (n - 1)d
Formula for the sum of the first n terms of an arithmetic sequence?
Sₙ = (n/2)(a₁ + aₙ)
nth term of a geometric sequence with initial term a₁ and ratio r?
aₙ = a₁ · r^(n - 1)
Sum of the first n terms of a finite geometric series?
Sₙ = a₁ (1 - rⁿ)/(1 - r), for r ≠ 1
Formula for the sum of an infinite geometric series (|r|<1)?
S = a₁ / (1 - r)
Write the general form of a polynomial equation in x of degree n.
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ = 0
Formula for factoring a quadratic x² + bx + c?
Find two numbers p, q such that p + q = b and pq = c, then x² + bx + c = (x + p)(x + q).
What is the formula for ‘completing the square’ on x² + bx?
x² + bx + (b/2)² = (x + b/2)²
Explain the general factoring formula for ‘difference of squares’ a² - b².
a² - b² = (a - b)(a + b)
Explain the factoring formula for ‘sum of cubes’ a³ + b³.
a³ + b³ = (a + b)(a² - ab + b²)
Explain the factoring formula for ‘difference of cubes’ a³ - b³.
a³ - b³ = (a - b)(a² + ab + b²)
State the formula for the axis of symmetry in a quadratic y = ax² + bx + c.
x = -b/(2a)
Write the power rule for exponents (aᵐ)ⁿ.
(aᵐ)ⁿ = a^(m·n)
Write the product rule for exponents aᵐ · aⁿ.
aᵐ · aⁿ = a^(m + n)
Write the quotient rule for exponents aᵐ / aⁿ, with a ≠ 0.
aᵐ / aⁿ = a^(m - n)
State the zero exponent rule for a ≠ 0.
a⁰ = 1
What is the negative exponent rule a^(-n), a ≠ 0?
a^(-n) = 1/(aⁿ)
Formula for converting a fractional exponent x^(p/q) to radicals?
x^(p/q) = √[q]{xᵖ}
What is the general form for a direct variation?
y = kx, where k is a constant
What is the general form for an inverse variation?
y = k/x, where k is a constant
Write the property of multiplication over addition (distributive property).
a(b + c) = ab + ac
Formula for the sum and difference product identity: (a + b)(a - b)?
a² - b²
Define absolute value inequality formula for |x| < a (a > 0).
-a < x < a
Define absolute value inequality formula for |x| > a (a > 0).
x < -a or x > a
Formula for the vertex (h, k) of a parabola y = ax² + bx + c?
h = -b/(2a), k = f(h)
In a fraction (rational expression) A/B, what must B satisfy?
B ≠ 0
Formula for adding two fractions p/q + r/s with no simplification assumed.
(ps + rq) / (qs)
Formula for multiplying two fractions p/q and r/s?
(p · r) / (q · s)
Formula for dividing p/q ÷ r/s (r≠0, s≠0)?
(p/q) × (s/r)
Express log form vs exponent form: y = logₐ x.
aʸ = x
Formula for slope m given parallel lines y = m₁x + b₁ and y = m₂x + b₂?
m₁ = m₂ if lines are parallel
Formula for slope if two lines y=m₁x +b₁ and y=m₂x +b₂ are perpendicular?
m₁ · m₂ = -1
Show how to rewrite x² + 2xy + y² in factored form.
(x + y)²
Show how to rewrite x² - y² in factored form.
(x - y)(x + y)
Formula to convert from slope-intercept form to standard form for y = mx + b?
mx - y + b = 0 (or multiply to clear fractions).
How do you solve for x in ax + b = 0?
x = -b/a, provided a≠0
Factor x² + 2xy + y² as a perfect square.
(x + y)²
Show the formula for the proportion property a/b = c/d → ad = bc.
Cross multiplication property
Formula for the sum of consecutive integers from 1 to n?
n(n+1)/2
Formula for an absolute value function’s vertex: y = a|x - h| + k?
Vertex is (h, k)
What is the standard form for a quadratic function?
ax² + bx + c, where a≠0
Write the factor form for a quadratic y = a(x - r₁)(x - r₂).
r₁, r₂ are the roots; expanded form is ax² + bx + c
