Algebra and Fundamentals - Cheat Sheet
Your last-minute revision companion for algebra essentials: exponents, radicals, logarithms, quadratics, progressions, and word problems. Every formula, every pitfall, every exam trigger condensed into one rapid-fire reference.
Exponents and Radicals
Formulas
a^m · a^n = a^(m+n)
Meaning
a = base; m, n = exponents
Watch out
Do NOT add bases: a^m + a^n ≠ a^(m+n). Only works for multiplication.
When to use
When multiplying powers with the same base
(a^m)^n = a^(mn)
Meaning
Nested exponents; a = base; m, n = exponents
Watch out
a^(m^n) ≠ (a^m)^n — parentheses matter; always compute m×n first.
When to use
Power raised to another power
(ab)^n = a^n · b^n
Meaning
Distribute exponent to each factor
Watch out
(a+b)^n ≠ a^n + b^n — this rule does NOT apply to sums.
When to use
Exponent applied to a product
a^(-n) = 1/a^n
Meaning
Negative exponent means reciprocal
Watch out
a^(-n) ≠ -a^n. Negative exponent flips; negative sign is different.
When to use
Simplifying expressions with negative powers
a^0 = 1 (a ≠ 0)
Meaning
Any nonzero base to the power 0 equals 1
Watch out
0^0 is indeterminate — never appears in exam if you're careful. a must be nonzero.
When to use
Simplification; elimination of terms
a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m
Meaning
Fractional exponent = nth root; m = power, n = root index
Watch out
Order matters: a^(m/n) ≠ a^(n/m). Always reduce m/n first if possible.
When to use
Converting between radical and exponential form
ⁿ√a = a^(1/n)
Meaning
nth root equals exponent 1/n
Watch out
ⁿ√(a+b) ≠ ⁿ√a + ⁿ√b — radicals do NOT distribute over sums.
When to use
Converting radicals to exponent form for simplification
ⁿ√(ab) = ⁿ√a · ⁿ√b
Meaning
nth root of product = product of nth roots
Watch out
Only works for products. ⁿ√(a/b) = ⁿ√a / ⁿ√b (same rule for division).
When to use
Simplifying radicals by factoring
ⁿ√(a^m) = a^(m/n)
Meaning
Radical with power exponent
Watch out
Simplify m/n before converting: e.g., ⁶√(a⁴) = a^(4/6) = a^(2/3) = ³√(a²).
When to use
Simplifying nested radicals and fractional exponents
Common values
Value
≈ 1.414
Symbol
√2
Quantity
√2
Value
≈ 1.732
Symbol
√3
Quantity
√3
Value
≈ 2.236
Symbol
√5
Quantity
√5
Value
√2 ≈ 1.414
Symbol
2^0.5
Quantity
2^(1/2)
Value
≈ 1.260
Symbol
³√2
Quantity
Cube root of 2
Important facts
- a^m / a^n = a^(m-n) (quotient rule: subtract exponents)
- √a · √b = √(ab) for a, b ≥ 0 (product rule for square roots)
- √a / √b = √(a/b) for b ≠ 0 (quotient rule for radicals)
- Rationalize denominators: multiply by the conjugate (e.g., 1/√2 × √2/√2 = √2/2)
- ⁿ√(a^n) = |a| for even n; = a for odd n (absolute value needed for even roots of negatives)
- Laws hold for any real base and exponent (when defined); negative bases with fractional exponents can yield complex numbers
Key definitions
Exponent (Power)
Example
In 2³ = 8, the exponent 3 means 2 × 2 × 2.
Number of times a base is multiplied by itself; in a^n, n is the exponent.
Radical
Example
√16 = 4; ³√8 = 2.
Expression written as ⁿ√a; the inverse of raising to a power; ⁿ√a = a^(1/n).
Rational Exponent
Example
x^(3/2) = (√x)³ = x√x.
Exponent expressed as a fraction m/n, where a^(m/n) = (ⁿ√a)^m.
Diagrams to know
- Law of exponents table (multiplication → add, division → subtract, power → multiply)
- Radical simplification tree (nested roots collapse via addition of exponent fractions)
Logarithms
Formulas
log_b(x) = y ⟺ b^y = x
Meaning
b = base, x = argument, y = logarithm; logarithm is the exponent
Watch out
b > 0, b ≠ 1, and x > 0 (logarithm undefined for x ≤ 0 or invalid base).
When to use
Converting between logarithmic and exponential form
log(MN) = log(M) + log(N)
Meaning
Logarithm of a product = sum of logarithms
Watch out
log(M) + log(N) ≠ log(M + N). Product rule, NOT sum rule.
When to use
Expanding log of products; solving equations
log(M/N) = log(M) − log(N)
Meaning
Logarithm of a quotient = difference of logarithms
Watch out
Applies to division ONLY. log(M − N) ≠ log(M) − log(N).
When to use
Expanding log of fractions
log(M^p) = p·log(M)
Meaning
Power moves out front; p can be any real number
Watch out
p·log(M) ≠ log(p) + log(M). The power becomes a coefficient, not a separate term.
When to use
Simplifying log of powers or extracting exponents
log_b(b) = 1
Meaning
Logarithm of the base in its own system equals 1
Watch out
log_10(10) = 1, ln(e) = 1. Each base has its own identity.
When to use
Simplification; recognizing identity
log_b(1) = 0
Meaning
Logarithm of 1 in any base is 0 (since b^0 = 1)
Watch out
Always true regardless of base (as long as base is valid).
When to use
Simplification; solving log equations
log_b(x) = ln(x) / ln(b) = log(x) / log(b)
Meaning
Change of base formula; convert any logarithm to natural or common log
Watch out
The NEW base (denominator) must be what you're converting TO. Never flip numerator/denominator.
When to use
Evaluating logarithms on calculators; switching bases
b^(log_b(x)) = x
Meaning
Exponential and logarithm are inverses; cancellation rule
Watch out
Only valid when base b and argument x are compatible (b > 0, b ≠ 1, x > 0).
When to use
Solving log and exponential equations
log_b(b^x) = x
Meaning
Logarithm of base to a power = the exponent
Watch out
Does NOT apply to log_b(a^x) where a ≠ b.
When to use
Inverse property; simplification
Common values
Value
≈ 0.301
Symbol
log(2)
Quantity
log₁₀(2)
Value
≈ 0.693
Symbol
ln(2)
Quantity
ln(2)
Value
= 1
Symbol
ln(e)
Quantity
ln(e)
Value
= 1
Symbol
log(10)
Quantity
log₁₀(10)
Value
≈ 2.718
Symbol
e
Quantity
e (Euler's number)
Important facts
- Logarithmic function is the inverse of exponential function: if y = b^x, then x = log_b(y)
- Domain of log_b(x): x > 0; range: all real numbers
- Base must satisfy b > 0 and b ≠ 1 for logarithm to be defined
- log(0) is undefined; log of negative numbers yields complex results in real domain
- ln(e) = 1, log(10) = 1 (identity in respective bases)
- Logarithmic equations often require checking solutions (argument must remain positive)
- ln ≈ 0.693 (natural log of 2); log₁₀(2) ≈ 0.301
Key definitions
Logarithm
Example
log₂(8) = 3 because 2³ = 8.
The exponent y such that b^y = x, written as log_b(x) = y.
Common Logarithm
Example
log(100) = 2 because 10² = 100.
Logarithm with base 10, written as log(x) or log₁₀(x).
Natural Logarithm
Example
ln(e) = 1; ln(e²) = 2.
Logarithm with base e ≈ 2.718, written as ln(x) or log_e(x).
Change of Base
Example
log₃(9) = ln(9) / ln(3) = 2.197 / 1.099 = 2.
Formula to convert logarithm from one base to another: log_b(x) = ln(x) / ln(b).
Diagrams to know
- Graph of y = log_b(x) (approaches −∞ as x → 0⁺, passes through (1,0), increases without bound)
- Comparison of bases: log_b(x) for b > 1 vs 0 < b < 1 (increasing vs decreasing curves)
Reactions or equations
log_b(x) = y ⟺ x = b^y
Note
Logarithm and exponential forms are equivalent; converting between them is key to solving equations.
Conditions
b > 0, b ≠ 1, x > 0
Quadratic and Polynomial Equations
Formulas
x = (−b ± √(b² − 4ac)) / (2a)
Meaning
a, b, c = coefficients of ax² + bx + c = 0; quadratic formula gives both roots
Watch out
Discriminant (b² − 4ac) sign determines root type: >0 (real distinct), =0 (repeated), <0 (complex). Always use ± to get both roots.
When to use
Solving any quadratic equation when factoring is difficult
Δ = b² − 4ac
Meaning
Discriminant; determines nature of quadratic roots
Watch out
Δ < 0 means NO real roots (only complex conjugates). Δ = 0 means roots are equal/repeated.
When to use
Analyzing root type without solving; checking if real solutions exist
Sum of roots = −b/a
Meaning
If roots are r₁ and r₂, then r₁ + r₂ = −b/a
Watch out
Sign: SUM is −b/a, not +b/a. Watch the coefficient signs in the original equation.
When to use
Finding one root if other is known; checking answers
Product of roots = c/a
Meaning
If roots are r₁ and r₂, then r₁ · r₂ = c/a
Watch out
If c and a have opposite signs, product is negative (one positive, one negative root).
When to use
Finding one root if other is known; analyzing root signs
(x − r₁)(x − r₂) = 0 where roots are r₁, r₂
Meaning
Factored form of quadratic; expansion gives ax² + bx + c = 0
Watch out
This is the factored form only if roots are integers or simple fractions. Use quadratic formula first if roots are irrational.
When to use
Writing quadratic when roots are known
x² − (sum)x + (product) = 0
Meaning
Quadratic in terms of sum and product of roots
Watch out
Remember the signs: −(sum) and +(product).
When to use
Building a quadratic when sum and product are given
For ax² + bx + c = 0 with real coefficients: if one root is p + qi, the other is p − qi
Meaning
Complex roots of polynomials with real coefficients always occur in conjugate pairs
Watch out
True only for REAL coefficients. If coefficients are complex, this rule does not hold.
When to use
Analyzing quadratics with negative discriminant
Important facts
- If a = 1, quadratic is monic: x² + bx + c = 0
- Quadratic always has exactly 2 roots (real or complex, counting multiplicity)
- If roots are integers and product/sum are integers, try factoring first (faster than quadratic formula)
- Complex roots of real quadratics always come in conjugate pairs: if a + bi is a root, so is a − bi
- Vertex form: y = a(x − h)² + k; vertex at (h, k); axis of symmetry x = h
- For ax² + bx + c = 0, vertex x-coordinate: x = −b / (2a); this is also the line of symmetry
- Graph opens upward if a > 0, downward if a < 0
Key definitions
Quadratic Equation
Example
2x² − 5x + 3 = 0; roots are x = 1 and x = 3/2.
Polynomial equation of degree 2: ax² + bx + c = 0, where a ≠ 0.
Discriminant
Example
For x² − 5x + 6 = 0: Δ = 25 − 24 = 1 > 0 → two real distinct roots.
Expression Δ = b² − 4ac; determines nature (real/complex) and number of roots.
Vieta's Formulas
Example
For x² − 5x + 6 = 0: sum = 5, product = 6; roots satisfy these via (3+2=5, 3×2=6).
Relations between roots and coefficients: sum = −b/a, product = c/a.
Repeated Root (Double Root)
Example
x² − 4x + 4 = (x − 2)² = 0 has double root x = 2.
When Δ = 0, the quadratic has one root with multiplicity 2; graph touches x-axis at one point.
Diagrams to know
- Parabola with two real roots (Δ > 0), one real repeated root (Δ = 0), no real roots (Δ < 0)
- Vertex, axis of symmetry, focus, and directrix of parabola y = ax² + bx + c
Reactions or equations
ax² + bx + c = a(x − r₁)(x − r₂)
Note
Factored form allows immediate identification of roots; expand to verify.
Conditions
r₁, r₂ are the roots of the quadratic
x² + px + q = (x + p/2)² − (p²/4 − q)
Note
Transforms to vertex form; reveals vertex and aids in solving.
Conditions
Completing the square; general technique
Arithmetic Progression (AP)
Formulas
a_n = a₁ + (n − 1)d
Meaning
a_n = nth term; a₁ = first term; d = common difference; n = term number
Watch out
n starts at 1, not 0. Off-by-one error is common: double-check which term you need.
When to use
Finding a specific term in an arithmetic sequence
d = a_n − a_(n−1) = a₂ − a₁
Meaning
Common difference; constant difference between consecutive terms
Watch out
d can be positive (increasing), negative (decreasing), or zero (constant sequence).
When to use
Identifying or verifying an AP; computing d from given terms
S_n = (n/2)[2a₁ + (n − 1)d]
Meaning
Sum of first n terms; n = number of terms; a₁ = first term; d = common difference
Watch out
Common error: forgetting the (n−1)d term or using wrong parentheses. Test with n=1 (should equal a₁).
When to use
Computing sum of first n terms of an AP
S_n = (n/2)(a₁ + a_n)
Meaning
Alternative sum formula using first and last terms
Watch out
Only works if you know the last term a_n. Verify n is correct.
When to use
When both a₁ and a_n are known (often simpler than other form)
a_n = a_m + (n − m)d
Meaning
Relating any two terms; a_m and a_n at positions m and n
Watch out
Must identify both m and n correctly from problem statement.
When to use
Finding term at position n given term at position m
2a_n = a_(n−1) + a_(n+1)
Meaning
Each term is the arithmetic mean of its neighbors
Watch out
This is the AP characterization; if ANY term fails this, sequence is not arithmetic.
When to use
Verifying AP property; finding unknown term in sequence
Important facts
- An AP is uniquely determined by any two distinct terms (can solve for a₁ and d)
- Sum S_n grows quadratically with n for d ≠ 0; linearly for d = 0
- If d > 0, AP increases; d < 0, AP decreases; d = 0, all terms equal
- For finite AP, the sum can also be computed as S_n = n × (median term), where median = (a₁ + a_n)/2
- The last term in sum formula is a_n = a₁ + (n−1)d, not a_(n+1)
- Inserting m arithmetic means between two numbers a and b creates AP with a+1 terms total and d = (b−a)/(m+1)
Key definitions
Arithmetic Progression (AP)
Example
3, 7, 11, 15, ... has a₁ = 3, d = 4; the 5th term is 3 + 4(4) = 19.
Sequence where consecutive terms have a constant difference d; a_n = a₁ + (n−1)d.
Common Difference
Example
In 10, 7, 4, 1, ..., d = −3 (decreasing AP).
Constant d such that each term is d more (or less) than the previous term.
Arithmetic Mean
Example
In 2, 5, 8, the mean of 2 and 8 is (2+8)/2 = 5.
Middle term of three consecutive AP terms; a_n = (a_(n−1) + a_(n+1)) / 2.
Diagrams to know
- AP sequence on a number line showing equal spacing
- Linear graph of term vs. term number (a_n vs n) — should be a straight line
Geometric Progression (GP)
Formulas
a_n = a₁ · r^(n−1)
Meaning
a_n = nth term; a₁ = first term; r = common ratio; n = term number
Watch out
n starts at 1. If r = 1, all terms equal a₁ (degenerate case). If r = 0, only a₁ ≠ 0.
When to use
Finding a specific term in a geometric sequence
r = a_n / a_(n−1) = a₂ / a₁
Meaning
Common ratio; ratio of any term to the previous term
Watch out
r can be positive, negative, or even complex. If |r| > 1, terms grow in magnitude; |r| < 1, terms decay.
When to use
Identifying or verifying a GP; computing r from given terms
S_n = a₁(r^n − 1) / (r − 1) for r ≠ 1
Meaning
Sum of first n terms; a₁ = first term; r = common ratio; n = number of terms
Watch out
Only valid for r ≠ 1. If r = 1, use S_n = n·a₁. Sign errors are common: (r^n − 1) not (1 − r^n) in numerator.
When to use
Computing sum of first n terms of a GP
S_n = a₁(1 − r^n) / (1 − r) for r ≠ 1
Meaning
Alternative form of GP sum (algebraically equivalent to above)
Watch out
Equivalent to standard form; use whichever feels natural. Both are correct.
When to use
When you prefer factor out (1 − r) in denominator; often clearer for |r| < 1
S_∞ = a₁ / (1 − r) for |r| < 1
Meaning
Sum of infinite GP; only converges if |r| < 1
Watch out
CRITICAL: If |r| ≥ 1, series diverges (sum is not finite). Must check |r| < 1 first.
When to use
Finding sum of infinite geometric series with |r| < 1
a_n = a_m · r^(n−m)
Meaning
Relating any two terms; a_m and a_n at positions m and n
Watch out
If m > n, exponent is negative; handle carefully.
When to use
Finding term at position n given term at position m
a_n² = a_(n−1) · a_(n+1)
Meaning
Each term is the geometric mean of its neighbors
Watch out
This is the GP characterization (for positive terms). If sequence has mixed signs, be careful with square roots.
When to use
Verifying GP property; finding unknown term in sequence
Important facts
- A GP is uniquely determined by any two distinct terms (can solve for a₁ and r via simultaneous equations)
- For |r| > 1, terms grow without bound; for |r| < 1, terms approach 0; for r = 1, all terms equal a₁
- Infinite sum S_∞ exists ONLY if |r| < 1; otherwise series diverges
- If r < 0, terms alternate in sign (e.g., 1, −2, 4, −8, ...)
- GP with r = 0 is degenerate: a₁, 0, 0, 0, ... (only first term matters)
- For finding n terms to reach a sum close to S_∞, check convergence: as n → ∞, r^n → 0 when |r| < 1
Key definitions
Geometric Progression (GP)
Example
2, 6, 18, 54, ... has a₁ = 2, r = 3; the 5th term is 2·3⁴ = 162.
Sequence where consecutive terms have a constant ratio r; a_n = a₁·r^(n−1).
Common Ratio
Example
In 100, 50, 25, 12.5, ..., r = 0.5 (decreasing GP).
Constant r such that each term is r times the previous term.
Geometric Mean
Example
In 1, 3, 9, the geometric mean of 1 and 9 is √(1×9) = 3.
Middle term of three consecutive GP terms; a_n = √(a_(n−1) · a_(n+1)).
Convergent Series
Example
1/2 + 1/4 + 1/8 + ... = (1/2)/(1 − 1/2) = 1.
Infinite GP with |r| < 1; sum converges to finite value S_∞ = a₁/(1−r).
Diagrams to know
- GP sequence on logarithmic scale (appears linear if plotted as log(a_n) vs n for r > 0)
- Exponential graph for r > 1 (growth), exponential decay for 0 < r < 1
Binomial Theorem
Formulas
(a + b)^n = Σ(k=0 to n) C(n,k) · a^(n−k) · b^k
Meaning
Sum of all binomial terms; C(n,k) = n!/(k!(n−k)!); k ranges 0 to n
Watch out
The exponents of a and b must sum to n at each term. Off-by-one errors in indexing are common.
When to use
Expanding (a+b)^n for any positive integer n
C(n,k) = n! / (k!(n−k)!)
Meaning
Binomial coefficient; number of ways to choose k items from n
Watch out
C(n,k) = C(n, n−k) (symmetry). 0! = 1, not 0.
When to use
Computing the coefficient of the kth term in binomial expansion
Term (r+1) = C(n,r) · a^(n−r) · b^r
Meaning
The (r+1)th term; r = 0, 1, 2, ..., n; note: r+1 because we start counting from term 1
Watch out
The rth term (counting from 1) uses C(n, r−1), not C(n, r). Always clarify: 'find the kth term' vs 'find the term with r'.
When to use
Finding a specific term without expanding the entire binomial
Coefficient of a^p in (a+b)^n is C(n, n−p) · b^p
Meaning
Extracting coefficient of a specific power of a
Watch out
The exponent sum (n−p) + p = n must hold. Verify power of a is achievable.
When to use
Finding coefficient of a desired term when a and b are specified
(1 + x)^n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ...
Meaning
Special case a=1, b=x; series form useful for fractional or negative n
Watch out
For |x| small and n not a positive integer, this is an infinite series, NOT finite. Convergence depends on |x|.
When to use
Expanding (1+x)^n or deriving series approximations
Common values
Value
= 1
Symbol
Always 1
Quantity
C(n, 0)
Value
= n
Symbol
Linear in n
Quantity
C(n, 1)
Value
= 1
Symbol
Always 1
Quantity
C(n, n)
Value
= n(n−1)/2
Symbol
Triangular numbers
Quantity
C(n, 2)
Important facts
- Binomial expansion has n+1 terms (from k=0 to k=n)
- Sum of all binomial coefficients in row n: C(n,0) + C(n,1) + ... + C(n,n) = 2^n
- The expansion is symmetric: coefficient of a^k equals coefficient of a^(n−k)
- For (a−b)^n, alternate signs: odd k terms are negative
- Factorial: n! = n(n−1)(n−2)...1; 0! = 1 by definition
- For large n, middle terms (k near n/2) have the largest coefficients
- Binomial theorem for negative or fractional n yields infinite series (use (1+x)^n form)
Key definitions
Binomial Theorem
Example
(x+2)³ = C(3,0)x³ + C(3,1)x²(2) + C(3,2)x(4) + C(3,3)(8) = x³ + 6x² + 12x + 8.
Formula for expanding (a+b)^n into a sum of n+1 terms, each with binomial coefficient C(n,k).
Binomial Coefficient
Example
C(5,2) = 5!/(2!3!) = 10.
Symbol C(n,k) or ⁿC_k; equals n!/(k!(n−k)!); represents coefficient in binomial expansion.
Pascal's Triangle
Example
Row 3: 1, 3, 3, 1 → coefficients for (a+b)³.
Triangular array of binomial coefficients; each row n gives coefficients for (a+b)^n.
Diagrams to know
- Pascal's Triangle (rows 0–10) showing binomial coefficients
- Graph of C(n,k) vs k for fixed n (bell-shaped for large n)
Reactions or equations
(a+b)^n − (a−b)^n = 2[b-terms with odd powers]
Note
Difference eliminates even-power terms; useful for extracting odd terms.
Conditions
Subtracting expansions with + and − in middle
(a+b)^n + (a−b)^n = 2[terms with even powers of b]
Note
Sum eliminates odd-power terms; useful for extracting even terms.
Conditions
Adding expansions with + and −
Word Problems – Work, Mixture, Age, and Motion
Formulas
Rate of work = 1 / (time to complete job) [jobs/time]
Meaning
Fraction of job completed per unit time
Watch out
ADD rates, NOT times. If A takes 6 days, B takes 4 days: combined rate = 1/6 + 1/4 = 5/12 jobs/day, not 1/(6+4).
When to use
Work-rate problems; multiple workers
Combined rate × time = 1 (job); t = 1 / (combined rate)
Meaning
Time to complete job together = 1 / (sum of individual rates)
Watch out
Only valid if working simultaneously on the SAME job. If they do different parts, use different equations.
When to use
Finding time for multiple workers to finish together
Amount = Concentration × Volume (or Quantity = Concentration × Amount of Solution)
Meaning
For mixtures: mass of solute = (concentration) × (total mass/volume)
Watch out
Keep units consistent. Concentration can be %, ratio, molarity, etc. — define before solving.
When to use
Mixture problems; tracking amounts of different components
Initial amount + Added − Removed = Final amount
Meaning
Conservation equation for mixture problems
Watch out
Sign convention: added is +, removed is −. Be careful with rates (e.g., drain removes per unit time).
When to use
Tracking total quantity when adding/removing components
Distance = Rate × Time; d = rt
Meaning
Fundamental motion equation
Watch out
Units must match: if d in km and t in hours, r must be km/h. Common error: mixing units.
When to use
Any motion problem; uniform or average velocity
Age problem: current age + years elapsed = future age
Meaning
Linear relationship for ages; set up equations comparing ages at different times
Watch out
All parties age at the same rate (1 year per year). Ratio of ages changes, but differences may stay constant.
When to use
Age relationship problems; comparing ages now vs. past/future
Important facts
- In work problems: total work = (combined rate) × (time); for multiple workers: 1 = (r₁ + r₂ + ...)t
- If a drain empties while pipes fill, subtract drain rate from total fill rate
- In mixture problems: (concentration₁)(volume₁) + (concentration₂)(volume₂) = (final concentration)(final volume)
- Average speed ≠ average of speeds; use total distance / total time
- In age problems, differences between ages remain constant over time, but ratios change
- For motion: if two objects move toward each other, add speeds; if same direction, subtract
- Relative velocity: if object A moves at v_A and B at v_B, A's velocity relative to B is v_A − v_B
Key definitions
Work Rate
Example
If a job takes 5 hours, the rate is 1/5 job/hour.
Fraction of a job completed per unit time; inverse of time to complete the full job.
Mixture Problem
Example
Mix 30% salt solution with 10% salt solution to get 20 L of 20% solution.
Problem involving combining solutions of different concentrations or materials of different values/types.
Age Problem
Example
Father is 3 times son's age now; in 10 years, he will be twice the son's age. Find current ages.
Problem comparing ages of two or more people at different times (now, past, or future).
Diagrams to know
- Work problem timeline (parallel work vs. sequential work)
- Mixture diagram (two inputs, different concentrations, one output)
- Motion diagram (distance-time or position-time for multiple objects)
Must remember
Quadratic formula: x = (−b ± √(b²−4ac))/(2a); discriminant Δ = b²−4ac determines root type (>0 real, =0 double, <0 complex).
AP sum: S_n = (n/2)[2a₁+(n−1)d] or (n/2)(a₁+a_n); GP sum: S_n = a₁(r^n−1)/(r−1) [r≠1]; infinite GP: S_∞ = a₁/(1−r) ONLY if |r|<1.
Laws of exponents: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^0 = 1. Radicals: ⁿ√(ab) = ⁿ√a · ⁿ√b (products only).
Logarithm rules: log(MN) = log(M)+log(N), log(M/N) = log(M)−log(N), log(M^p) = p·log(M). Change base: log_b(x) = ln(x)/ln(b).
Sum and product of quadratic roots: sum = −b/a, product = c/a (Vieta's formulas). Use to verify answers or build quadratic from roots.
Binomial coefficient: C(n,k) = n!/(k!(n−k)!); (r+1)th term of (a+b)^n is C(n,r)·a^(n−r)·b^r. Sum of row: Σ C(n,k) = 2^n.
Work problems: ADD RATES, NOT TIMES. If A finishes in t₁ and B in t₂, combined rate = 1/t₁ + 1/t₂; time together = 1/(rate sum).
GP convergence: S_∞ exists ONLY if |r|<1. If r=1, sum = n·a₁; if |r|>1 or |r|=1 (r≠1), series diverges.
Inverse: log_b(x) = y ⟺ b^y = x. Check domains: b>0, b≠1, x>0 for real logarithms. b^(log_b(x)) = x and log_b(b^x) = x (inverse properties).
Common mistakes: (a+b)^n ≠ a^n+b^n (power does NOT distribute), log(M+N) ≠ log(M)+log(N) (log is NOT linear), ⁿ√(a+b) ≠ ⁿ√a + ⁿ√b.
Last minute tips
Quadratic formula is your safety net: if factoring fails or seems slow, use x = (−b±√(b²−4ac))/(2a) immediately. Check discriminant sign FIRST to know if real roots exist.
Rationale
Exam time pressure makes factoring risky; formula is mechanical and reliable. Discriminant check saves wasted work on complex-root problems.
In work problems, ALWAYS ask: 'Am I adding rates or times?' If workers work simultaneously, add rates (1/t₁ + 1/t₂); if sequentially, add times. Exam loves this confusion.
Rationale
This is the #1 algebraic mistake on word-problem questions. One sentence in your mind prevents the wrong setup.
For infinite GP sum: FIRST check |r|<1. If you skip this check, you'll get a wrong 'answer' from S_∞ = a₁/(1−r) on a divergent series. Mark it DIVERGES if |r|≥1.
Rationale
Exam graders love this trap. One mental check ('Is |r| less than 1?') prevents automatic point loss.
When expanding (a+b)^n with binomial theorem: write out the PATTERN of exponents first (a^n, a^(n−1)b, a^(n−2)b², ..., b^n) before computing coefficients. This avoids off-by-one errors.
Rationale
Most errors come from coefficient confusion, not formula. Visual template reduces careless mistakes under time pressure.
For logarithm problems: convert to exponential form (log_b(x)=y → b^y=x) if stuck. Often reveals the answer or simplifies algebra dramatically.
Rationale
Logarithmic form can feel abstract; exponent form is concrete and more intuitive. Conversion is a powerful troubleshooting tool.
Comparison tables
Exponents vs. Radicals: Quick Reference
| Operation | Exponent Form | Radical Form | Rule/Property |
|---|---|---|---|
| Power of a product | (ab)^n | Not applicable | = a^n · b^n |
| Product of powers | a^m · a^n | Not applicable | = a^(m+n) |
| Power of a power | (a^m)^n | Not applicable | = a^(mn) |
| Radical to exponent | a^(m/n) | ⁿ√(a^m) | equivalent forms |
| Product of radicals | a^(1/n) · b^(1/n) | ⁿ√a · ⁿ√b | = ⁿ√(ab) |
| Quotient of radicals | a^(1/n) / b^(1/n) | ⁿ√a / ⁿ√b | = ⁿ√(a/b) |
AP vs. GP: Side-by-Side Comparison
| Property | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
| General term | a_n = a₁ + (n−1)d | a_n = a₁ · r^(n−1) |
| Constant between consecutive terms | Common difference d (addition) | Common ratio r (multiplication) |
| Sum of first n terms | S_n = (n/2)[2a₁ + (n−1)d] = (n/2)(a₁ + a_n) | S_n = a₁(r^n − 1)/(r − 1) [r ≠ 1] |
| Infinite sum | Diverges unless d = 0 (converges to a₁) | S_∞ = a₁/(1−r) only if |r| < 1 |
| Middle term relationship | 2a_n = a_(n−1) + a_(n+1) (arithmetic mean) | a_n² = a_(n−1) · a_(n+1) (geometric mean) |
| Sum of two given terms | If a₁ and d known: a_m + a_n = a₁(2 + (m+n−2)d/1) | If a₁ and r known: a_m · a_n = a₁² · r^(m+n−2) |
| Graph shape | Linear (a_n vs n is a straight line) | Exponential (a_n vs n curves up/down depending on r) |
Logarithm Rules Summary
| Rule Name | Formula | Example | Watch Out |
|---|---|---|---|
| Product rule | log_b(MN) = log_b(M) + log_b(N) | log₁₀(2×5) = log₁₀(2) + log₁₀(5) = log₁₀(10) = 1 | Does NOT apply to sums: log(M+N) ≠ log(M) + log(N) |
| Quotient rule | log_b(M/N) = log_b(M) − log_b(N) | ln(e²/e) = 2 − 1 = 1 | Only for division; log(M−N) ≠ log(M) − log(N) |
| Power rule | log_b(M^p) = p · log_b(M) | log₂(8³) = 3 log₂(8) = 3(3) = 9 | Power becomes a coefficient, not separate term |
| Change of base | log_b(x) = log_c(x) / log_c(b) | log₃(9) = ln(9) / ln(3) = 2.197 / 1.099 ≈ 2 | New base (denominator) is what you're converting TO |
| Inverse property | b^(log_b(x)) = x; log_b(b^x) = x | 2^(log₂(5)) = 5; log₃(3^7) = 7 | Exponent and log must have same base |
Root Types: Discriminant (Δ) Analysis
| Discriminant Δ | Sign of Δ | Root Type | Number of Real Roots | Example |
|---|---|---|---|---|
| Δ = b² − 4ac | Δ > 0 | Two distinct real roots | 2 | x² − 5x + 6 = 0; Δ = 1; roots: 2, 3 |
| Δ = 0 | Δ = 0 | One repeated real root (double root) | 1 (with multiplicity 2) | x² − 4x + 4 = 0; Δ = 0; root: x = 2 (double) |
| Δ < 0 | Δ < 0 | Two complex conjugate roots | 0 (in reals) | x² + x + 1 = 0; Δ = −3; roots: (−1 ± i√3)/2 |
Word Problem Strategy: Problem Type vs. Method
| Problem Type | Key Equation(s) | Variable Setup | Common Pitfall |
|---|---|---|---|
| Work (parallel workers) | r₁ + r₂ = combined rate; t = 1/(r₁+r₂) | r₁ = 1/t₁, r₂ = 1/t₂ | Adding TIMES (not rates) — WRONG. |
| Mixture (two solutions) | c₁V₁ + c₂V₂ = c_f(V₁+V₂) | c = concentration, V = volume, subscript f = final | Forgetting units; mixing % and decimal concentration |
| Motion (same direction) | d = v₁t ± v₂t = (v₁ ± v₂)t | d = distance, v = velocity, t = time | Using average speed incorrectly; mixing units (km vs m) |
| Age (comparing times) | Age_now + Δt = Age_future; set up ratio at past/present/future | Assign variables to current ages; work backward/forward | Forgetting that all parties age at 1 year/year |
Diagrams
Algebra Fundamentals: Master Concept Map
Master concept map showing hierarchical relationships between all major algebra topics: exponents/radicals, logarithms, quadratics, progressions, binomial theorem, and word-problem types. Use this to quickly recall which formula family applies to a given problem.
Quadratic Equation Solution Decision Tree
Step-by-step flowchart to guide you through solving quadratics: first attempt factoring, then check discriminant to understand root type, finally apply quadratic formula if needed. Emphasizes verification via Vieta's formulas.