Explore our comprehensive range of mathematics subjects, each designed to build strong foundations and develop advanced problem-solving skills.
Core mathematics covering essential concepts and problem-solving techniques
Equations, inequalities, and graphical representations
Solving equations of the form ax + b = c and representing them graphically
Solving equations of the form ax² + bx + c = 0 using factorization and formula
Solving systems of linear and non-linear equations
Working with higher-degree polynomials and their properties
Solve the quadratic equation: 2x² - 5x - 3 = 0
Step 1: Identify a = 2, b = -5, c = -3
Step 2: Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Step 3: x = (5 ± √(25 + 24)) / 4
Step 4: x = (5 ± √49) / 4 = (5 ± 7) / 4
Solution: x = 3 or x = -0.5
Algebra Essentials
Comprehensive guide to core algebraic concepts
Khan Academy
Free online lessons and practice problems
Shapes, angles, and trigonometric relationships
Properties of 2D and 3D shapes, angles, and transformations
Understanding sine, cosine, tangent and their applications
Angles in circles, tangents, and chord properties
Vector operations and geometric transformations
Find the area of a triangle with sides 5cm, 7cm, and 10cm.
Step 1: Use Heron's formula: Area = √(s(s-a)(s-b)(s-c))
Step 2: Calculate s = (a+b+c)/2 = (5+7+10)/2 = 11
Step 3: Area = √(11(11-5)(11-7)(11-10))
Step 4: Area = √(11 × 6 × 4 × 1) = √264
Solution: Area ≈ 16.25 cm²
Geometry Fundamentals
Clear explanations of geometric concepts
GeoGebra
Interactive geometry software for visualization
Differentiation, integration, and vector operations
Rules of differentiation, gradients, and rates of change
Definite and indefinite integrals, areas under curves
Vector operations, dot and cross products
Optimization, kinematics, and area problems
Find the derivative of f(x) = 3x⁴ - 2x² + 5x - 7
Step 1: Apply the power rule: d/dx(xⁿ) = n·xⁿ⁻¹
Step 2: d/dx(3x⁴) = 3·4x³ = 12x³
Step 3: d/dx(-2x²) = -2·2x = -4x
Step 4: d/dx(5x) = 5
Step 5: d/dx(-7) = 0
Solution: f'(x) = 12x³ - 4x + 5
Calculus Made Easy
Step-by-step guide to calculus concepts
Desmos Calculator
Online graphing tool for visualization
Advanced mathematical concepts for higher-level study and university preparation
Understanding and working with numbers in the form a + bi
Addition, subtraction, multiplication, and division of complex numbers
Geometric representation of complex numbers
Powers and roots of complex numbers
Solving equations with complex coefficients and solutions
Express (3 + 4i)(2 - 5i) in the form a + bi.
Step 1: Use FOIL method: (3 + 4i)(2 - 5i) = 3(2) + 3(-5i) + 4i(2) + 4i(-5i)
Step 2: = 6 - 15i + 8i - 20i²
Step 3: Since i² = -1, we have: 6 - 15i + 8i - 20(-1)
Step 4: = 6 - 15i + 8i + 20
Solution: = 26 - 7i
Complex Analysis
Comprehensive guide to complex numbers
Complex Visualizer
Interactive tool for visualizing complex numbers
Complex numbers are essential for many university courses, including:
Equations involving derivatives and their solutions
Separable and linear first-order differential equations
Homogeneous and non-homogeneous equations with constant coefficients
Modeling real-world phenomena with differential equations
Analytical and numerical approaches to solving DEs
Solve the differential equation: dy/dx = 2x + y with initial condition y(0) = 1
Step 1: Rearrange to standard form: dy/dx - y = 2x
Step 2: Identify as a first-order linear equation
Step 3: Use integrating factor method with e^(-∫dx) = e^(-x)
Step 4: Multiply both sides: e^(-x)dy/dx - e^(-x)y = 2xe^(-x)
Step 5: Recognize left side as derivative of product: d/dx(e^(-x)y) = 2xe^(-x)
Step 6: Integrate: e^(-x)y = ∫2xe^(-x)dx = -2xe^(-x) - 2e^(-x) + C
Step 7: Solve for y: y = -2x - 2 + Ce^x
Step 8: Apply initial condition y(0) = 1: 1 = -2(0) - 2 + C(1)
Step 9: Solve for C: C = 3
Solution: y = -2x - 2 + 3e^x
Differential Equations
Comprehensive textbook with examples
ODE Solver
Online tool for solving and visualizing DEs
Differential equations are fundamental in:
Matrix operations, determinants, and linear transformations
Addition, subtraction, multiplication, and inversion of matrices
Calculating and applying determinants
Finding and applying eigenvalues and eigenvectors
Representing and composing transformations using matrices
Find the determinant of the matrix:
A = [ 3 1 2 ]
[ 4 -2 5 ]
[ 1 0 3 ]
Step 1: Use the formula for 3×3 determinant:
det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)
Step 2: Substitute values:
det(A) = 3[(-2)(3) - (5)(0)] - 1[(4)(3) - (5)(1)] + 2[(4)(0) - (-2)(1)]
Step 3: Simplify:
det(A) = 3(-6) - 1(12 - 5) + 2(0 + 2)
Step 4: Calculate:
det(A) = -18 - 7 + 4 = -21
Solution: det(A) = -21
Linear Algebra and Its Applications
Standard reference text for linear algebra
Matrix Calculator
Online tool for matrix operations
Linear algebra is crucial for:
Data analysis, probability theory, and statistical inference
Fundamentals of probability theory and statistical distributions
Probability laws, conditional probability, and independence
Binomial, Poisson, and geometric distributions
Normal, exponential, and uniform distributions
Central limit theorem and sampling theory
A fair coin is tossed 8 times. What is the probability of getting exactly 5 heads?
Step 1: This is a binomial probability problem with n = 8 trials, k = 5 successes, and p = 0.5 probability of success
Step 2: Use the binomial probability formula: P(X = k) = (n choose k) × p^k × (1-p)^(n-k)
Step 3: Calculate (8 choose 5) = 8!/(5!(8-5)!) = 8!/(5!3!) = 56
Step 4: P(X = 5) = 56 × (0.5)^5 × (0.5)^3
Step 5: P(X = 5) = 56 × (0.5)^8 = 56 × 0.00390625 = 0.21875
Solution: The probability is 0.21875 or approximately 21.9%
Introduction to Probability
Comprehensive guide to probability theory
StatKey
Interactive tool for exploring distributions
Probability and distributions are used in:
Statistical inference and decision-making based on data
Formulating and interpreting statistical hypotheses
Testing means and proportions
Testing for independence and goodness of fit
Analysis of variance for comparing multiple groups
A company claims that its new process reduces the mean production time to less than 25 minutes. A sample of 36 production times has a mean of 23.5 minutes with a standard deviation of 4.2 minutes. Test this claim at the 5% significance level.
Step 1: Set up hypotheses: H₀: μ ≥ 25, H₁: μ < 25
Step 2: Calculate the test statistic: z = (x̄ - μ₀)/(σ/√n)
Step 3: z = (23.5 - 25)/(4.2/√36) = -1.5/0.7 = -2.14
Step 4: For α = 0.05 in a one-tailed test, the critical value is -1.645
Step 5: Since -2.14 < -1.645, we reject the null hypothesis
Conclusion: There is sufficient evidence to support the claim that the new process reduces the mean production time to less than 25 minutes.
Statistical Inference
Comprehensive guide to hypothesis testing
StatCrunch
Online statistical software for data analysis
Recommended tools for statistical analysis:
Modeling relationships between variables
Modeling the relationship between two variables
Models with multiple predictor variables
Measuring the strength of relationships
Assessing model assumptions and fit
A researcher collected data on hours studied (x) and exam scores (y) for 10 students:
| Hours Studied (x) | Exam Score (y) |
|---|---|
| 1 | 65 |
| 2 | 70 |
| 3 | 75 |
| 4 | 80 |
| 5 | 85 |
Find the linear regression equation and predict the exam score for a student who studies 6 hours.
Step 1: Calculate the means: x̄ = 3, ȳ = 75
Step 2: Calculate the slope: b = Σ(x-x̄)(y-ȳ)/Σ(x-x̄)² = 50/10 = 5
Step 3: Calculate the y-intercept: a = ȳ - b·x̄ = 75 - 5(3) = 60
Step 4: The regression equation is: y = 60 + 5x
Step 5: For x = 6: y = 60 + 5(6) = 90
Solution: The predicted exam score for a student who studies 6 hours is 90.
Applied Regression Analysis
Comprehensive guide to regression methods
Regression Visualizer
Interactive tool for exploring regression models
Recommended tools for regression analysis: