Algebra, Core Formulas and Identities
Algebraic formulas are the building blocks of all higher mathematics. Master these core concepts to unlock success in advanced courses.
Essential Equation Forms
The quadratic formula solves ax2 + bx + c = 0 with x = (-b ± sqrt(b2 - 4ac)) / 2a. The discriminant (b2 - 4ac) determines root types: positive gives two real roots, zero gives one repeated root, and negative gives two complex conjugate roots.
Slope-intercept form (y = mx + b) expresses lines clearly. Here, m is slope and b is the y-intercept. Calculate slope as (y2-y1)/(x2-x1). Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes.
Point-slope form (y - y1 = m(x - x1)) is useful when you know a point and slope but not the y-intercept.
Distance and Coordinate Geometry
The distance formula d = sqrt((x2-x1)2 + (y2-y1)2) comes from the Pythagorean theorem. Use it to find distances between two points. The midpoint formula is ((x1+x2)/2, (y1+y2)/2).
Exponents and Logarithms
Master these exponent rules:
- Product rule: a^m * a^n = a^(m+n)
- Quotient rule: a^m / a^n = a^(m-n)
- Power rule: (a^m)^n = a^(mn)
- Zero exponent: a^0 = 1
- Negative exponent: a^(-n) = 1/a^n
- Fractional exponent: a^(m/n) = nth root of a^m
Logarithm rules include log(xy) = log(x) + log(y), log(x/y) = log(x) - log(y), and log(x^n) = n(log(x)). Use change of base as log_b(x) = ln(x)/ln(b).
Factoring and Polynomials
Key factoring patterns:
- Difference of squares: a2 - b2 = (a+b)(a-b)
- Perfect square trinomial: a2 + 2ab + b2 = (a+b)2
- Sum of cubes: a3 + b3 = (a+b)(a2 - ab + b2)
- Difference of cubes: a3 - b3 = (a-b)(a2 + ab + b2)
Use completing the square to rewrite ax2 + bx + c as a(x-h)2 + k. This reveals the vertex form with vertex at (h, k).
Absolute Value and Inequalities
Absolute value |x| measures distance from zero. |x| = x if x >= 0, and |x| = -x if x < 0. Solve |x| = a as x = a or x = -a. For |x| < a, use -a < x < a. For |x| > a, use x > a or x < -a.
Reverse the inequality sign when multiplying or dividing by a negative number. For quadratic inequalities, find roots and test intervals.
Systems, Sequences, and Functions
Solve systems of equations by substitution, elimination, or graphing. Solutions are either one solution (independent), no solution (parallel lines), or infinitely many (same line).
Arithmetic sequences have constant difference d. Find the nth term as a_n = a_1 + (n-1)d. Sum is S_n = n(a_1 + a_n)/2.
Geometric sequences have constant ratio r. Find the nth term as a_n = a_1(r^(n-1)). Sum is S_n = a_1(1-r^n)/(1-r). For infinite series with |r| < 1, use S = a_1/(1-r).
The binomial theorem expands (a+b)^n as sum of C(n,k)(a^(n-k))(b^k). Find coefficients C(n,k) = n!/(k!(n-k)!) using Pascal's Triangle.
Functions have domain (valid inputs) and range (outputs). Restrictions occur when denominators equal zero, even roots have negative radicands, or logarithm arguments are non-positive. Use the vertical line test to verify functions. Find inverses only for one-to-one functions.
| Term | Meaning |
|---|---|
| Quadratic Formula | For ax2 + bx + c = 0: x = (-b +/- sqrt(b2 - 4ac)) / 2a. The discriminant (b2 - 4ac) determines roots: positive = two real, zero = one repeated, negative = two complex conjugate. |
| Slope-Intercept Form | y = mx + b. m = slope, b = y-intercept. Slope = (y2-y1)/(x2-x1). Parallel: equal slopes. Perpendicular: negative reciprocals. |
| Point-Slope Form | y - y1 = m(x - x1). Useful when you know a point and slope but not the y-intercept. |
| Distance Formula | d = sqrt((x2-x1)2 + (y2-y1)2). Derived from Pythagorean theorem. Midpoint: ((x1+x2)/2, (y1+y2)/2). |
| Exponent Rules | Product: a^m * a^n = a^(m+n). Quotient: a^m/a^n = a^(m-n). Power: (a^m)^n = a^(mn). Zero: a^0 = 1. Negative: a^(-n) = 1/a^n. Fractional: a^(m/n) = nth root of a^m. |
| Logarithm Rules | log(xy) = log(x)+log(y). log(x/y) = log(x)-log(y). log(x^n) = n*log(x). Change of base: log_b(x) = ln(x)/ln(b). |
| Factoring Patterns | Difference of squares: a2-b2 = (a+b)(a-b). Perfect square: a2+2ab+b2 = (a+b)2. Sum of cubes: a3+b3 = (a+b)(a2-ab+b2). Difference of cubes: a3-b3 = (a-b)(a2+ab+b2). |
| Absolute Value | |x| = x if x>=0, -x if x<0. Distance from zero. |x|=a gives x=a or x=-a. |x|<a gives -a<x<a. |x|>a gives x>a or x<-a. |
| Systems of Equations | Solve by substitution, elimination, or graphing. One solution (independent), no solution (parallel), infinitely many (same line). |
| Arithmetic Sequences | Constant difference d. nth term: a_n = a_1 + (n-1)d. Sum: S_n = n(a_1+a_n)/2. |
| Geometric Sequences | Constant ratio r. nth term: a_n = a_1*r^(n-1). Sum: S_n = a_1(1-r^n)/(1-r). Infinite (|r|<1): S = a_1/(1-r). |
| Binomial Theorem | (a+b)^n = sum of C(n,k)*a^(n-k)*b^k. C(n,k) = n!/(k!(n-k)!). Pascal's Triangle gives coefficients. |
| Polynomial Division | Long or synthetic division. Remainder Theorem: f(c) = remainder when divided by (x-c). Factor Theorem: (x-c) is factor if f(c)=0. |
| Completing the Square | Rewrite ax2+bx+c as a(x-h)2+k. Vertex form of parabola, vertex at (h,k). Used to derive quadratic formula. |
| Inequalities | Reverse sign when multiplying/dividing by negative. Compound: and=intersection, or=union. Quadratic: find roots, test intervals. |
| Functions, Domain and Range | Domain: valid inputs. Range: outputs. Restrictions: denominator!=0, even root radicand>=0, log argument>0. Vertical line test. Composition: f(g(x)). Inverse: one-to-one only. |
Geometry and Trigonometry
Essential formulas for standardized tests, physics, and calculus help you solve geometry and trigonometry problems quickly.
Foundational Theorems
The Pythagorean theorem states a2 + b2 = c2. Common triples include (3,4,5), (5,12,13), (8,15,17), and (7,24,25). Memorizing these saves time on exams.
Area and Volume Formulas
Calculate area using:
- Triangle: (1/2)bh
- Rectangle: lw
- Circle: pi(r2)
- Trapezoid: (1/2)(b1 + b2)h
- Using Heron's formula: sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter
Find volume with:
- Rectangular prism: lwh
- Cylinder: pi(r2)h
- Cone: (1/3)pi(r2)h
- Sphere: (4/3)pi(r3)
- Pyramid: (1/3)Bh (where B is base area)
Circle Properties
Circumference is C = 2(pi)r. Arc length is s = r(theta) in radians. An inscribed angle equals half the central angle. The circle equation is (x-h)2 + (y-k)2 = r2.
Trigonometric Ratios and Identities
Remember SOH-CAH-TOA: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent. The reciprocals are cosecant, secant, and cotangent.
On the unit circle at key angles:
- 0 degrees: sin = 0, cos = 1
- 30 degrees (pi/6): sin = 1/2, cos = sqrt(3)/2
- 45 degrees (pi/4): sin = cos = sqrt(2)/2
- 60 degrees (pi/3): sin = sqrt(3)/2, cos = 1/2
- 90 degrees (pi/2): sin = 1, cos = 0
Pythagorean identities include sin2 + cos2 = 1, tan2 + 1 = sec2, and 1 + cot2 = csc2.
Solving Triangles
The law of sines uses a/sinA = b/sinB = c/sinC for AAS, ASA, and SSA cases.
The law of cosines states c2 = a2 + b2 - 2ab(cosC). Use this for SAS or SSS cases. It generalizes the Pythagorean theorem.
Double angle formulas are sin(2x) = 2(sinx)(cosx) and cos(2x) = cos2x - sin2x = 2cos2x - 1 = 1 - 2sin2x.
Angle addition formulas include sin(A ± B) = sinA(cosB) ± cosA(sinB) and cos(A ± B) = cosA(cosB) ∓ sinA(sinB).
Coordinate and Advanced Geometry
Convert between radians and degrees: pi radians equals 180 degrees. Multiply degrees by pi/180 to get radians. Multiply radians by 180/pi to get degrees.
Similar triangles have equal angles (AA, SAS, or SSS similarity). With scale factor k, sides have ratio k, areas have ratio k2, and volumes have ratio k3.
Conic sections include circle (x-h)2 + (y-k)2 = r2, ellipse x2/a2 + y2/b2 = 1, parabola y = a(x-h)2 + k, and hyperbola x2/a2 - y2/b2 = 1.
Vectors v = (a, b) have magnitude sqrt(a2 + b2). The dot product u·v = a1(a2) + b1(b2) = |u||v|cos(theta). Perpendicular vectors have dot product zero.
Trigonometric graphs follow y = A(sin(Bx+C)) + D. Amplitude is |A|, period is 2pi/|B|, phase shift is -C/B, and vertical shift is D.
| Term | Meaning |
|---|---|
| Pythagorean Theorem | a2+b2=c2. Common triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). |
| Area Formulas | Triangle: (1/2)bh. Rectangle: lw. Circle: pi*r2. Trapezoid: (1/2)(b1+b2)h. Heron's: sqrt(s(s-a)(s-b)(s-c)). |
| Volume Formulas | Prism: lwh. Cylinder: pi*r2*h. Cone: (1/3)pi*r2*h. Sphere: (4/3)pi*r3. Pyramid: (1/3)Bh. |
| Circle Theorems | C = 2*pi*r. Arc length: s = r*theta. Inscribed angle = (1/2) central angle. Equation: (x-h)2+(y-k)2=r2. |
| SOH-CAH-TOA | sin = opp/hyp. cos = adj/hyp. tan = opp/adj. Reciprocals: csc=1/sin, sec=1/cos, cot=1/tan. |
| Unit Circle, Key Angles | 0: sin=0,cos=1. 30(pi/6): sin=1/2,cos=sqrt3/2. 45(pi/4): sin=cos=sqrt2/2. 60(pi/3): sin=sqrt3/2,cos=1/2. 90(pi/2): sin=1,cos=0. |
| Pythagorean Identities | sin2+cos2=1. tan2+1=sec2. 1+cot2=csc2. |
| Law of Sines | a/sinA = b/sinB = c/sinC. For AAS, ASA, SSA (ambiguous case). |
| Law of Cosines | c2 = a2+b2-2ab*cosC. For SAS or SSS. Generalizes Pythagorean theorem. |
| Double Angle Formulas | sin(2x)=2sinx*cosx. cos(2x)=cos2x-sin2x=2cos2x-1=1-2sin2x. |
| Angle Addition Formulas | sin(A+/-B) = sinA*cosB +/- cosA*sinB. cos(A+/-B) = cosA*cosB -/+ sinA*sinB. |
| Radian-Degree Conversion | pi rad = 180 deg. Deg to rad: multiply by pi/180. Rad to deg: multiply by 180/pi. |
| Similar Triangles | Equal angles (AA, SAS, SSS similarity). Scale factor k: sides ratio k, areas ratio k2, volumes ratio k3. |
| Conic Sections | Circle: (x-h)2+(y-k)2=r2. Ellipse: x2/a2+y2/b2=1. Parabola: y=a(x-h)2+k. Hyperbola: x2/a2-y2/b2=1. |
| Vectors | v=<a,b>. Magnitude: sqrt(a2+b2). Dot product: u*v=a1a2+b1b2=|u||v|cos(theta). Perpendicular: dot product=0. |
| Trigonometric Graphs | y=A*sin(Bx+C)+D. Amplitude=|A|. Period=2pi/|B|. Phase shift=-C/B. Vertical shift=D. |
Calculus, Derivatives, Integrals, and Limits
Quick recall of differentiation and integration rules lets you focus on problem-solving strategy instead of formula derivation.
Limits and Continuity
The definition of a limit states lim(x→a) f(x) = L. A limit exists only if the left limit equals the right limit. Key limits include lim(x→0) sin(x)/x = 1 and lim(x→∞) (1+1/x)^x = e.
Continuity at point a requires: f(a) is defined, the limit exists, and the limit equals f(a). The intermediate value theorem states that if f is continuous on [a, b] and k is between f(a) and f(b), then some c exists where f(c) = k.
Derivatives and Differentiation
The derivative definition is f'(x) = lim(h→0) [f(x+h)-f(x)]/h. It represents instantaneous rate of change and tangent line slope.
Apply these basic differentiation rules:
- Power rule: d/dx(x^n) = nx^(n-1)
- Product rule: d/dx(f·g) = f'g + fg'
- Quotient rule: d/dx(f/g) = (f'g - fg')/g2
- Chain rule: d/dx(f(g(x))) = f'(g(x))·g'(x)
Trigonometric derivatives are:
- d/dx(sin) = cos
- d/dx(cos) = -sin
- d/dx(tan) = sec2
- d/dx(sec) = sec(tan)
- d/dx(arcsin) = 1/sqrt(1-x2)
- d/dx(arctan) = 1/(1+x2)
Exponential and logarithmic derivatives:
- d/dx(e^x) = e^x
- d/dx(a^x) = a^x(ln(a))
- d/dx(ln(x)) = 1/x
Integration and the Fundamental Theorem
The fundamental theorem of calculus has two parts. Part 1: d/dx(integral from a to x of f(t)dt) = f(x). Part 2: integral from a to b of f(x)dx = F(b) - F(a).
Apply basic integration formulas:
- x^n: x^(n+1)/(n+1) + C
- 1/x: ln|x| + C
- e^x: e^x + C
- sin: -cos + C
- cos: sin + C
- sec2: tan + C
Use u-substitution by choosing u = g(x), calculating du = g'(x)dx, rewriting the integral, integrating, and substituting back.
Integration by parts uses integral(u·dv) = uv - integral(v·du). Use the LIATE priority for selecting u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.
Advanced Techniques and Theorems
L'Hopital's Rule applies when lim(f/g) is 0/0 or infinity/infinity. Then lim(f/g) = lim(f'/g') if the limit exists.
The mean value theorem states that if f is continuous on [a, b] and differentiable on (a, b), then some c exists where f'(c) = (f(b)-f(a))/(b-a).
The definite integral as area: integral from a to b of f(x)dx equals net signed area. Area above the x-axis is positive, below is negative. For area between curves, use integral([f-g]dx).
Related rates problems require differentiating an equation relating quantities with respect to time. Substitute known rates and solve for the unknown.
Taylor and Maclaurin series approximate functions:
- e^x = sum(x^n/n!)
- sin(x) = sum((-1)^n·x^(2n+1)/(2n+1)!)
- cos(x) = sum((-1)^n·x^(2n)/(2n)!)
- 1/(1-x) = sum(x^n)
Optimization requires finding critical points where f'(x) = 0 or is undefined. Use the second derivative test or first derivative test to identify maxima and minima. Always check endpoints for closed intervals. Express the quantity you want to optimize as a function of one variable.
| Term | Meaning |
|---|---|
| Definition of a Limit | lim(x->a) f(x) = L. Exists only if left = right limit. Key: lim sin(x)/x = 1, lim (1+1/x)^x = e. |
| Derivative Definition | f'(x) = lim(h->0) [f(x+h)-f(x)]/h. Instantaneous rate of change and tangent slope. |
| Basic Differentiation Rules | Power: nx^(n-1). Product: f'g+fg'. Quotient: (f'g-fg')/g2. Chain: f'(g(x))*g'(x). |
| Trig Derivatives | d/dx sin=cos. d/dx cos=-sin. d/dx tan=sec2. d/dx sec=sec*tan. d/dx arcsin=1/sqrt(1-x2). d/dx arctan=1/(1+x2). |
| Exp/Log Derivatives | d/dx e^x=e^x. d/dx a^x=a^x*ln(a). d/dx ln(x)=1/x. |
| Fundamental Theorem of Calculus | Part 1: d/dx integral(a to x) f(t)dt = f(x). Part 2: integral(a to b) f(x)dx = F(b)-F(a). |
| Basic Integration Rules | x^n: x^(n+1)/(n+1)+C. 1/x: ln|x|+C. e^x: e^x+C. sin: -cos+C. cos: sin+C. sec2: tan+C. |
| U-Substitution | Choose u=g(x), du=g'(x)dx, rewrite, integrate, substitute back. |
| Integration by Parts | integral u*dv = uv - integral v*du. LIATE priority for u: Log, Inverse trig, Algebraic, Trig, Exponential. |
| L'Hopital's Rule | If lim f/g is 0/0 or inf/inf, then lim f/g = lim f'/g' (if exists). |
| Mean Value Theorem | Continuous on [a,b], differentiable on (a,b): exists c where f'(c) = (f(b)-f(a))/(b-a). |
| Definite Integral as Area | integral(a to b) f(x)dx = net signed area. Above x-axis positive, below negative. Between curves: integral [f-g]dx. |
| Related Rates | Differentiate equation relating quantities with respect to time. Substitute known rates, solve for unknown. |
| Taylor/Maclaurin Series | e^x = sum x^n/n!. sinx = sum (-1)^n*x^(2n+1)/(2n+1)!. cosx = sum (-1)^n*x^(2n)/(2n)!. 1/(1-x) = sum x^n. |
| Continuity | f continuous at a: f(a) defined, lim exists, lim=f(a). IVT: continuous on [a,b], k between f(a) and f(b), exists c with f(c)=k. |
| Optimization | Find critical points (f'=0 or undefined). Test with second derivative or first derivative test. Check endpoints for closed intervals. Express quantity to optimize as function of one variable. |
How to Study math Effectively
Mastering math requires the right study approach, not just more hours. Research in cognitive science consistently shows that three techniques produce the best learning outcomes: active recall, spaced repetition, and interleaving.
Active recall means testing yourself rather than re-reading. Spaced repetition reviews material at scientifically-optimized intervals. Interleaving mixes related topics rather than studying one in isolation. FluentFlash is built around all three.
When you study math with our FSRS algorithm, every term is scheduled for review at exactly the moment you're about to forget it. This maximizes retention while minimizing study time.
Why Passive Review Fails
The most common mistake students make is relying on passive review methods. Re-reading your notes, highlighting textbook passages, or watching lecture videos feels productive. However, studies show these methods produce only 10-20% of the retention that active recall achieves.
Flashcards force your brain to retrieve information, which strengthens memory pathways far more than recognition alone. Pair this with spaced repetition scheduling, and you can learn in 20 minutes a day what would take hours of passive review.
Your Practical Study Plan
Start by creating 15-25 flashcards covering the highest-priority concepts. Review them daily for the first week using our FSRS scheduling. As cards become easier, intervals automatically expand from minutes to days to weeks. You're always working on material at the edge of your knowledge.
After 2-3 weeks of consistent practice, math concepts become automatic rather than effortful to recall.
Study Steps to Follow
- Generate flashcards using FluentFlash AI or create them manually from your notes
- Study 15-20 new cards per day, plus scheduled reviews
- Use multiple study modes (flip, multiple choice, written) to strengthen recall
- Track your progress and identify weak topics for focused review
- Review consistently. Daily practice beats marathon sessions
- 1
Generate flashcards using FluentFlash AI or create them manually from your notes
- 2
Study 15-20 new cards per day, plus scheduled reviews
- 3
Use multiple study modes (flip, multiple choice, written) to strengthen recall
- 4
Track your progress and identify weak topics for focused review
- 5
Review consistently, daily practice beats marathon sessions
Why Flashcards Work Better Than Other Study Methods for math
Flashcards aren't just for vocabulary. They're one of the most research-backed study tools for any subject, including math. The reason comes down to how memory works.
When you read a textbook passage, your brain stores that information in short-term memory. Without retrieval practice, it fades within hours. Flashcards force retrieval, which is the mechanism that transfers information from short-term to long-term memory.
The Testing Effect
The testing effect, documented in hundreds of peer-reviewed studies, shows that students who study with flashcards consistently outperform those who re-read by 30-60% on delayed tests. This isn't because flashcards contain more information. It's because retrieval strengthens neural pathways in a way that passive exposure cannot.
Every time you successfully recall a math concept from a flashcard, you're making that concept easier to recall next time.
How FSRS Amplifies Results
FluentFlash amplifies this effect with the FSRS algorithm, a modern spaced repetition system that schedules reviews at mathematically-optimal intervals based on your actual performance. Cards you find easy get pushed further into the future. Cards you struggle with come back sooner.
Over time, this builds remarkable retention with minimal time investment. Students using FSRS-based systems typically retain 85-95% of material after 30 days, compared to roughly 20% retention from passive review alone.