Wednesday, 16 April 2025

25 Essential formula that every student should know

25 Essential Math Formulas Every Student Should Know

25 Essential Math Formulas Every Student Should Know (With Examples)

Whether you're preparing for exams or just brushing up your math skills, these formulas are your go-to toolkit. Here's a categorized list with easy-to-follow examples.

1. Algebra

1. Quadratic Formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Example: For \( ax^2 + bx + c = 0 \), solve using the formula above.

2. Difference of Squares:

\[ a^2 - b^2 = (a - b)(a + b) \]

Example: \( 16 - 9 = (4 - 3)(4 + 3) = 7 \)

3. (a + b)2 Expansion:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

Example: \( (2 + 3)^2 = 4 + 12 + 9 = 25 \)

2. Geometry

4. Area of Triangle:

\[ A = \frac{1}{2}bh \]

Example: base = 5, height = 4 → Area = \( \frac{1}{2} \times 5 \times 4 = 10 \)

5. Area of Circle:

\[ A = \pi r^2 \]

Example: r = 3 → Area = \( \pi \times 9 \approx 28.27 \)

6. Circumference of Circle:

\[ C = 2\pi r \]

Example: r = 3 → Circumference = \( 2 \times \pi \times 3 \approx 18.85 \)

3. Trigonometry

7. Pythagorean Theorem:

\[ a^2 + b^2 = c^2 \]

Example: a = 3, b = 4 → \( c = 5 \)

8. Sine Rule:

For any triangle ABC, with sides a, b, and c opposite to angles A, B, and C respectively:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

9. Cosine Rule:

\[ c^2 = a^2 + b^2 - 2ab \cos C \]

10. Basic Trig Ratios:

\[ \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}}, \cos \theta = \frac{\text{base}}{\text{hypotenuse}}, \tan \theta = \frac{\text{perpendicular}}{\text{base}} \]

4. Calculus

11. Derivative of xn:

\[ \frac{d}{dx}x^n = nx^{n-1} \]

12. Derivative of sin x:

\[ \frac{d}{dx}\sin x = \cos x \]

13. Derivative of ex:

\[ \frac{d}{dx}e^x = e^x \]

14. Integral of xn:

\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \ (n \ne -1) \]

15. Integral of 1/x:

\[ \int \frac{1}{x} dx = \ln |x| + C \]

5. Probability and Statistics

16. Mean:

\[ \text{Mean} = \frac{\sum x}{n} \]

17. Median:

\[ \text{Middle value when data is sorted} \]

18. Mode:

\[ \text{Most frequently occurring value in the data set} \]

19. Permutations:

\[ ^nP_r = \frac{n!}{(n-r)!} \]

20. Combinations:

\[ ^nC_r = \frac{n!}{r!(n-r)!} \]

6. Miscellaneous

21. Sum of n Natural Numbers:

\[ \sum_{i=1}^n i = \frac{n(n+1)}{2} \]

22. Sum of Squares:

\[ \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \]

23. Sum of Cubes:

\[ \sum_{i=1}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2 \]

24. Distance Formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

25. Midpoint Formula:

\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Conclusion: Keep these formulas handy as you tackle math problems across various domains. Practice regularly to master their use!

Thank You for Reading! I hope you found this guide on essential math formulas helpful and informative. If you enjoyed this post, don't forget to bookmark it and share it with your friends or classmates. Your support helps this blog grow and allows me to continue sharing valuable content. Stay curious, keep practicing, and check back soon for more math tricks and tips!

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