Algebra Basic Flashcards
1
Q
Slope formula
A
m = (y₂ - y₁) / (x₂ - x₁)
2
Q
Quadratic formula
A
x = (-b ± √(b² - 4ac)) / 2a
3
Q
Distance formula
A
d = √((x₂ - x₁)² + (y₂ - y₁)²)
4
Q
Midpoint formula
A
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
5
Q
Pythagorean theorem
A
a² + b² = c²
6
Q
Slope-intercept form
A
y = mx + b
7
Q
Standard form of a linear equation
A
Ax + By = C
8
Q
Point-slope form
A
y - y₁ = m(x - x₁)
9
Q
Difference of squares formula
A
a² - b² = (a - b)(a + b)
10
Q
Factored form of a quadratic equation
A
a(x - r₁)(x - r₂)
11
Q
FOIL method
A
(a + b)(c + d) = ac + ad + bc + bd
12
Q
Vertex form of a quadratic equation
A
y = a(x - h)² + k
13
Q
Discriminant formula
A
b² - 4ac
14
Q
Equation of a circle
A
(x - h)² + (y - k)² = r²
15
Q
Exponential growth formula
A
A = A₀ e^(rt)
16
Q
Logarithm rule: logₐ(xy)
A
logₐ x + logₐ y
17
Q
Logarithm rule: logₐ(x/y)
A
logₐ x - logₐ y
18
Q
Logarithm rule: logₐ(xᵇ)
A
b logₐ x
19
Q
Sum of an arithmetic sequence
A
Sₙ = (n/2) (a + l)
20
Q
General form of a quadratic equation
A
ax² + bx + c = 0
21
Q
Sum of the roots of a quadratic equation
A
-b/a
22
Q
Product of the roots of a quadratic equation
A
c/a
23
Q
Equation of a line given two points (x₁, y₁) and (x₂, y₂)
A
y - y₁ = (y₂ - y₁) / (x₂ - x₁) * (x - x₁)
24
Q
Exponential decay formula
A
A = A₀ e^(-rt)
25
Compound interest formula
A = P(1 + r/n)^(nt)
26
Simple interest formula
I = PRT
27
Formula for the nth term of an arithmetic sequence
aₙ = a₁ + (n - 1)d
28
Formula for the nth term of a geometric sequence
aₙ = a₁ * r^(n-1)
29
Sum of a finite geometric series
Sₙ = a₁(1 - rⁿ) / (1 - r)
30
Sum of an infinite geometric series (|r| < 1)
S = a₁ / (1 - r)
31
Logarithmic identity: logₐ(1)
0
32
Logarithmic identity: logₐ(a)
1
33
Change of base formula
logₐ b = log_c b / log_c a
34
Power rule for exponents
(a^m)^n = a^(m*n)
35
Product rule for exponents
a^m * a^n = a^(m+n)
36
Quotient rule for exponents
a^m / a^n = a^(m-n)
37
Zero exponent rule
a⁰ = 1 (for a ≠ 0)
38
Negative exponent rule
a^(-n) = 1 / a^n
39
Radical to exponent conversion
√a = a^(1/2)
40
Variable definition
A symbol (often x) that represents an unknown or changeable value
41
Constant definition
A fixed value that does not change
42
Coefficient definition
A numerical factor multiplying a variable (e.g., 5 in 5x)
43
Term definition
A single mathematical expression (e.g., 3x, -7, 2y²)
44
Like terms definition
Terms with the same variables to the same powers (e.g., 2x and 5x)
45
Expression vs. Equation
Expression: A combination of terms (3x + 5); Equation: Two expressions set equal (3x + 5 = 14)
46
Distributive property formula
a(b + c) = ab + ac
47
Commutative property of addition
a + b = b + a
48
Associative property of multiplication
(ab)c = a(bc)
49
Additive inverse definition
The opposite of a number (a + (-a) = 0)
50
Combining like terms rule
Add/subtract coefficients of terms that have identical variable parts
51
Simplify 3x + 7 + 2x - 4
5x + 3
52
Factor out the greatest common factor (GCF) in 8x + 12
4(2x + 3)
53
Expand (x + 2)(x + 3)
x² + 5x + 6
54
Simplify 2(x + 4) - 3(x - 1)
-x + 11
55
Explain FOIL method
Multiply First terms, Outer terms, Inner terms, Last terms in two binomials
56
Rewrite 12x - 6y using factoring
6(2x - y)
57
Expression for perimeter of a rectangle (length x and width y)
P = 2x + 2y
58
Expression for area of a triangle (base b and height h)
A = (1/2)bh
59
Definition of polynomial
An expression of finite terms with nonnegative integer exponents (e.g., x² - 4x + 7)
60
Linear equation standard form
Ax + By = C
61
Slope-intercept form
y = mx + b
62
Slope formula between points (x₁, y₁) and (x₂, y₂)
m = (y₂ - y₁) / (x₂ - x₁)
63
Y-intercept definition
The point where a line crosses the y-axis (x=0)
64
Solve for x: 3x - 9 = 0
x = 3
65
Solve for x: x/2 + 2 = 5
x = 6
66
Inequality solution example: Solve x + 3 > 7
x > 4
67
Graph of y = x + 2
A straight line with slope 1 and y-intercept (0, 2)
68
Solve for y: -2y + 4 = 10
y = -3
69
Rewrite the inequality x < -2 in interval notation
(-∞, -2)
70
Definition of a system of equations
Two or more equations with the same variables
71
Solution to a system of equations
A set of variable values satisfying all equations simultaneously
72
Substitution method
Solve one equation for a variable, then substitute into the other equation
73
Elimination method
Add or subtract equations to eliminate a variable
74
Solve by substitution: x + y = 4 and y = 2x
x = 4/3; y = 8/3
75
Solve by elimination: 2x + y = 7, x - y = 1
x = 8/3; y = 5/3
76
Graphical method definition
Plot each line; the intersection point gives the solution
77
No solution to a system
Lines are parallel with different intercepts (inconsistent system)
78
Infinite solutions to a system
Lines are coincident (the same line)
79
Solve system: x + y = 2, x + y = 5
No solution (parallel lines)
80
Product rule of exponents
a^m · a^n = a^(m+n)
81
Quotient rule of exponents
a^m / a^n = a^(m-n) (a≠0)
82
Power rule of exponents
(a^m)^n = a^(mn)
83
Zero exponent rule
a^0 = 1 (a≠0)
84
Negative exponent rule
a^(-n) = 1 / a^n
85
Simplify: x² · x³
x⁵
86
Simplify: (x³)²
x⁶
87
Square root of x in radical form
√x = x^(1/2)
88
Cube root of x in radical form
∛x = x^(1/3)
89
Rational exponent to radical conversion
x^(m/n) = n√(x^m)
90
Sum of cubes factorization
a³ + b³ = (a + b)(a² - ab + b²)
91
Difference of cubes factorization
a³ - b³ = (a - b)(a² + ab + b²)
92
Difference of squares factorization
a² - b² = (a - b)(a + b)
93
Trinomial quadratic factorization example
x² + 5x + 6 = (x + 2)(x + 3)
94
Factor 3x² + 6x
3x(x + 2)
95
Factor 2x² - 8x + 6
2(x - 3)(x - 1)
96
Expand (x - 4)(x + 2)
x² - 2x - 8
97
Factor x² - 16
(x - 4)(x + 4)
98
Factor x² + x - 6
(x + 3)(x - 2)
99
Solve x² - 9 = 0 by factoring
(x - 3)(x + 3) = 0 → x=3 or x=-3
100
What is a variable in algebra?
A letter or symbol that represents an unknown or changeable value.
101
What is a constant in algebra?
A fixed value that does not change.
102
What is a coefficient?
A numerical factor multiplied by a variable (e.g., 7 in 7x).
103
Define a ‘term’ in an expression.
A single mathematical entity (e.g., 5, 3x, or -2y²).
104
What is an algebraic expression?
A combination of terms (variables, constants, operations) but no equals sign.
105
What distinguishes an equation from an expression?
An equation has an equals sign, an expression does not.
106
What is a polynomial?
An expression with one or more terms, each having nonnegative integer exponents on variables.
107
Define a monomial.
A polynomial with exactly one term (e.g., 6x²).
108
Define a binomial.
A polynomial with two distinct terms (e.g., x + 3).
109
Define a trinomial.
A polynomial with three distinct terms (e.g., x² + 5x + 6).
110
What is the degree of a polynomial?
The highest sum of exponents on any term (e.g., degree of 4x³y is 4).
111
Write the standard form of a polynomial in x.
List terms in descending powers of x (e.g., 3x³ - 2x² + x + 1).
112
Define a zero (root) of a polynomial.
A value of x that makes the polynomial equal to zero.
113
What is a linear equation in one variable?
An equation of the form ax + b = 0, where a ≠ 0.
114
What is the slope-intercept form of a line?
y = mx + b, where m is slope and b is y-intercept.
115
State the slope formula between (x₁, y₁) and (x₂, y₂).
m = (y₂ - y₁) / (x₂ - x₁).
116
What is the standard form of a line?
Ax + By = C.
117
What is the point-slope form of a line?
y - y₁ = m(x - x₁).
118
Define ‘slope’ in coordinate geometry.
The ratio of vertical change to horizontal change: rise/run.
119
What is a y-intercept?
The point where the graph crosses the y-axis (x=0).
120
What is an x-intercept?
The point where the graph crosses the x-axis (y=0).
121
Define ‘domain’ of a function f(x).
All possible x-values for which f(x) is defined.
122
Define ‘range’ of a function f(x).
All possible output values f(x) can take.
123
What is a function in algebra?
A relation where each input has exactly one output.
124
Explain the vertical line test in graphs.
If any vertical line intersects a graph more than once, it’s not a function.
125
What is meant by composite function (f ∘ g)(x)?
It means f(g(x)), applying g first, then f.
126
Define the process of ‘factoring’ in algebra.
Rewriting an expression as a product of simpler expressions.
127
What is the Greatest Common Factor (GCF)?
The largest expression that divides all terms of a polynomial.
128
Write the difference of squares formula.
a² - b² = (a - b)(a + b).
129
Write the sum of squares factorization.
a² + b² is not factorable over the reals (no simple factorization).
130
Write the sum of cubes formula.
a³ + b³ = (a + b)(a² - ab + b²).
131
Write the difference of cubes formula.
a³ - b³ = (a - b)(a² + ab + b²).
132
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.
133
Give an example of factoring a trinomial: x² + 5x + 6.
x² + 5x + 6 = (x + 2)(x + 3).
134
Define a prime polynomial (over the reals).
A polynomial that cannot be factored further (other than trivial factors).
135
What does irreducible polynomial mean?
Same as prime polynomial: it cannot be factored over a given number system.
136
Define a ‘root’ or solution to an equation.
A value that satisfies the equation, making it true.
137
What does ‘zero of a polynomial’ mean?
A value of x for which the polynomial equals 0 (same as root).
138
Define an ‘extraneous solution.’
A solution that emerges from algebraic manipulation but doesn’t satisfy the original equation.
139
What is a system of equations?
Two or more equations with the same variables.
140
Define a ‘consistent system’ of linear equations.
A system that has at least one solution.
141
Define an ‘inconsistent system’ of linear equations.
A system that has no solutions.
142
Define a ‘dependent system’ of linear equations.
A system with infinitely many solutions; equations represent the same line.
143
What does f(x) mean in algebra?
Function notation: f is the function’s name, x is the input variable.
144
What is composition of functions, like f(g(x))?
Applying g first, then plugging the result into f.
145
State the quadratic formula for ax² + bx + c = 0.
x = [-b ± √(b² - 4ac)] / (2a).
146
What is the discriminant in a quadratic equation?
b² - 4ac.
147
What is the sum of the roots of ax² + bx + c = 0?
−b/a.
148
What is the product of the roots of ax² + bx + c = 0?
c/a.
149
Factoring x² - 16 uses which formula?
Difference of squares: x² - 16 = (x - 4)(x + 4).
150
What is the FOIL method?
First, Outer, Inner, Last multiplication for binomials.
151
What is the Binomial Theorem for (a + b)ⁿ?
Σ from k=0 to n of [nCk · a^(n-k) · b^k].
152
Define a rational expression.
A fraction with polynomials in numerator and denominator.
153
Why must we check domain restrictions for rational expressions?
Because denominator cannot be zero.
154
How do you simplify a rational expression?
Factor numerator and denominator, then cancel common factors.
155
How to add two rational expressions with denominators D1 and D2?
Use the common denominator D1·D2 (or LCM if simpler), then add numerators.
156
How to multiply two rational expressions?
Multiply numerators, multiply denominators, then simplify.
157
How to divide two rational expressions?
Multiply by the reciprocal of the second expression.
158
Define a radical expression.
An expression involving a root symbol (√, ∛, etc.).
159
What is the ‘radicand’ in √x?
The value under the radical sign (x).
160
Define an nth root, e.g., √[n]{x}.
A number that when raised to the nth power gives x.
161
What does a rational exponent x^(m/n) mean?
The nth root of x^m, or (x^m)^(1/n).
162
State the property √a × √b = √(ab).
Product of square roots = square root of the product.
163
State the property √(a/b) = √a / √b (b≠0).
Quotient under a radical = quotient of radicals.
164
What is the absolute value of a number?
Its distance from zero on the real number line.
165
Define an inequality in algebra.
A statement showing <, >, ≤, or ≥ between expressions.
166
Solve x + 3 > 5 in simple form.
x > 2.
167
Write x > 2 in interval notation.
(2, ∞).
168
What is a compound inequality?
Two inequalities joined by ‘and’ or ‘or’ (e.g., 1 < x < 3).
169
Define ‘domain’ of an inequality solution.
The set of all real numbers that satisfy that inequality.
170
State the reciprocal property: x × (1/x).
It equals 1 for x ≠ 0.
171
What does a negative exponent mean? x^(-n).
1/(x^n), for x ≠ 0.
172
What does (ab)^n mean?
a^n × b^n.
173
How do you combine like terms in an expression?
Add/subtract coefficients of terms with identical variable parts.
174
What are ‘like terms’?
Terms with the same variable(s) raised to the same powers.
175
State the commutative property of addition.
a + b = b + a.
176
State the commutative property of multiplication.
ab = ba.
177
State the associative property of addition.
(a + b) + c = a + (b + c).
178
State the associative property of multiplication.
(ab)c = a(bc).
179
State the identity property of addition.
a + 0 = a.
180
State the identity property of multiplication.
a × 1 = a.
181
What is the additive inverse property?
a + (-a) = 0.
182
What is the multiplicative inverse property?
a × (1/a) = 1, for a ≠ 0.
183
Interpret a^m / a^n (a≠0).
a^(m-n).
184
Interpret (a^m)^n.
a^(mn).
185
When do we use polynomial long division?
To divide one polynomial by another of equal or lower degree.
186
When do we use synthetic division?
To divide a polynomial by a binomial of the form (x - c).
187
What is the direct variation formula?
y = kx, for some constant k.
188
What is the inverse variation formula?
y = k/x, for some constant k (x≠0).
189
What is a joint variation formula?
z = kxy, for some constant k.
190
Difference between ‘expression’ and ‘equation’ in usage?
Expression is a phrase of math terms; equation is a statement with an equals sign.
191
Why use slope-intercept form y=mx+b?
It lets us easily read slope (m) and y-intercept (b).
192
What is the purpose of standard form Ax + By = C?
Often used for integer coefficients and analyzing intercepts quickly.
193
How do we use the slope formula?
To find the rate of change between two points on a line.
194
Difference between a function and a relation?
A function has each input mapped to exactly one output; a relation can have multiple.
195
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.
196
What is the substitution method for solving a 2-variable system?
Solve one equation for one variable, then substitute into the other.
197
What is the elimination method for solving a 2-variable system?
Add or subtract equations to eliminate one variable, solve for the other.
198
How do you solve a linear system graphically?
Plot both lines; intersection point is the solution (if it exists).
199
How is factoring used to solve x² + bx + c = 0?
Factor into (x - r₁)(x - r₂)=0, so x=r₁ or x=r₂.
200
What is the general form of a linear equation in slope-intercept form?
y = mx + b
201
Write the formula for finding the x-intercept of a line y = mx + b.
Set y = 0, so x = -b/m
202
Write the formula for finding the y-intercept of a line Ax + By = C.
Set x = 0, then y = C/B
203
State the formula for the slope of a line in standard form Ax + By = C (B ≠ 0).
Slope m = -A/B
204
Give the formula for the slope of a line perpendicular to one with slope m.
Slope = -1/m
205
What is the formula for the line parallel to y = mx + b through (x₀, y₀)?
y - y₀ = m(x - x₀)
206
State the formula to convert a decimal r to a percentage.
Percentage = r × 100%
207
Give the point-slope form of a line through (x₁, y₁) with slope m.
y - y₁ = m(x - x₁)
208
Formula for an arithmetic sequence’s nth term?
aₙ = a₁ + (n - 1)d
209
Formula for the sum of the first n terms of an arithmetic sequence?
Sₙ = (n/2)(a₁ + aₙ)
210
nth term of a geometric sequence with initial term a₁ and ratio r?
aₙ = a₁ · r^(n - 1)
211
Sum of the first n terms of a finite geometric series?
Sₙ = a₁ (1 - rⁿ)/(1 - r), for r ≠ 1
212
Formula for the sum of an infinite geometric series (|r|<1)?
S = a₁ / (1 - r)
213
Write the general form of a polynomial equation in x of degree n.
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0
214
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).
215
What is the formula for ‘completing the square’ on x² + bx?
x² + bx + (b/2)² = (x + b/2)²
216
Explain the general factoring formula for ‘difference of squares’ a² - b².
a² - b² = (a - b)(a + b)
217
Explain the factoring formula for ‘sum of cubes’ a³ + b³.
a³ + b³ = (a + b)(a² - ab + b²)
218
Explain the factoring formula for ‘difference of cubes’ a³ - b³.
a³ - b³ = (a - b)(a² + ab + b²)
219
State the formula for the axis of symmetry in a quadratic y = ax² + bx + c.
x = -b/(2a)
220
Write the power rule for exponents (aᵐ)ⁿ.
(aᵐ)ⁿ = a^(m·n)
221
Write the product rule for exponents aᵐ · aⁿ.
aᵐ · aⁿ = a^(m + n)
222
Write the quotient rule for exponents aᵐ / aⁿ, with a ≠ 0.
aᵐ / aⁿ = a^(m - n)
223
State the zero exponent rule for a ≠ 0.
a⁰ = 1
224
What is the negative exponent rule a^(-n), a ≠ 0?
a^(-n) = 1/(aⁿ)
225
Formula for converting a fractional exponent x^(p/q) to radicals?
x^(p/q) = √[q]{xᵖ}
226
What is the general form for a direct variation?
y = kx, where k is a constant
227
What is the general form for an inverse variation?
y = k/x, where k is a constant
228
Write the property of multiplication over addition (distributive property).
a(b + c) = ab + ac
229
Formula for the sum and difference product identity: (a + b)(a - b)?
a² - b²
230
Define absolute value inequality formula for |x| < a (a > 0).
-a < x < a
231
Define absolute value inequality formula for |x| > a (a > 0).
x < -a or x > a
232
Formula for the vertex (h, k) of a parabola y = ax² + bx + c?
h = -b/(2a), k = f(h)
233
In a fraction (rational expression) A/B, what must B satisfy?
B ≠ 0
234
Formula for adding two fractions p/q + r/s with no simplification assumed.
(ps + rq) / (qs)
235
Formula for multiplying two fractions p/q and r/s?
(p · r) / (q · s)
236
Formula for dividing p/q ÷ r/s (r≠0, s≠0)?
(p/q) × (s/r)
237
Express log form vs exponent form: y = logₐ x.
aʸ = x
238
Formula for slope m given parallel lines y = m₁x + b₁ and y = m₂x + b₂?
m₁ = m₂ if lines are parallel
239
Formula for slope if two lines y=m₁x +b₁ and y=m₂x +b₂ are perpendicular?
m₁ · m₂ = -1
240
Show how to rewrite x² + 2xy + y² in factored form.
(x + y)²
241
Show how to rewrite x² - y² in factored form.
(x - y)(x + y)
242
Formula to convert from slope-intercept form to standard form for y = mx + b?
mx - y + b = 0 (or multiply to clear fractions).
243
How do you solve for x in ax + b = 0?
x = -b/a, provided a≠0
244
Factor x² + 2xy + y² as a perfect square.
(x + y)²
245
Show the formula for the proportion property a/b = c/d → ad = bc.
Cross multiplication property
246
Formula for the sum of consecutive integers from 1 to n?
n(n+1)/2
247
Formula for an absolute value function’s vertex: y = a|x - h| + k?
Vertex is (h, k)
248
What is the standard form for a quadratic function?
ax² + bx + c, where a≠0
249
Write the factor form for a quadratic y = a(x - r₁)(x - r₂).
r₁, r₂ are the roots; expanded form is ax² + bx + c
