A. Algebra Formulas

1. Quadratic Formula:   \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

2. Slope-Intercept Form of a Line:   \( y = mx + b \)

3. Distance Formula:   \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)

4. Midpoint Formula:   \( \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \)

5. Factoring Quadratic Expressions:   \( ax^2 + bx + c = 0 \)

   \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

6. Binomial Theorem:   \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

7. Arithmetic Progression (AP) Sum:   \( S_n = \frac{n}{2}[2a + (n-1)d] \)

8. Geometric Progression (GP) Sum:   \( S_n = \frac{a(r^n-1)}{r-1} \)

9. Quadratic Equations Sum/Product of Roots:   \( \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \)

10. Arithmetic Mean (Average): \( \text{Average} = \frac{\text{Sum of Values}}{\text{Number of Values}} \)

11. Geometric Mean:     \( GM = \sqrt[n]{a_1 \cdot a_2 \cdot \ldots \cdot a_n} \)

12. Quadratic Inequality:     \( ax^2 + bx + c > 0 \quad \text{or} \quad ax^2 + bx + c < 0 \)

13. Sum of Cubes:     \( a^3 + b^3 = (a + b)(a^2 – ab + b^2) \)

14. Difference of Cubes: \( a^3 – b^3 = (a – b)(a^2 + ab + b^2) \)

15. Arithmetic Mean-Geometric Mean Inequality: \( \frac{a_1 + a_2 + \ldots + a_n}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdot \ldots \cdot a_n} \)

16. Remainder Theorem:     \( \text{If } P(x) \text{ is divided by } (x – a), \text{ then the remainder is } P(a) \)

17. Factorial Notation:     \( n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \)

18. Combination Formula:     \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

19. Permutation Formula: \( P(n, r) = \frac{n!}{(n-r)!} \)

B. Geometry Formulas

1. Area of a Circle:     \( A = \pi r^2 \)

2. Circumference of a Circle:     \( C = 2\pi r \)

3. Area of a Triangle:     \( A = \frac{1}{2}bh \)

4. Area of a Rectangle:     \( A = lw \)

5. Volume of a Cylinder:     \( V = \pi r^2 h \)

6. Surface Area of a Sphere:     \( A = 4\pi r^2 \)

7. Pythagorean Theorem:     \( c^2 = a^2 + b^2 \)

8. Law of Sines:     \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

9. Law of Cosines:     \( c^2 = a^2 + b^2 – 2ab\cos(C) \)

10. Perimeter of a Square: \( P = 4s \)

11. Volume of a Cone: \( V = \frac{1}{3}\pi r^2 h \)

12. Sum of Interior Angles in a Polygon:     \( S = (n-2) \times 180^\circ \)

13. Area of a Trapezoid:     \( A = \frac{1}{2}(a+b)h \)

14. Area of a Parallelogram: \( A = bh \)

15. Area of a Rhombus:     \( A = \frac{d_1 \times d_2}{2} \)

16. Area of a Kite:     \( A = \frac{d_1 \times d_2}{2} \)

17. Volume of a Sphere: \( V = \frac{4}{3}\pi r^3 \)

18. Volume of a Pyramid:     \( V = \frac{1}{3}Bh \)

19. Surface Area of a Cone:     \( A = \pi r(r + \sqrt{h^2 + r^2}) \)

20. Area of a Sector:     \( A = \frac{\theta}{360^\circ} \pi r^2 \)

21. Law of Tangents:     \( \frac{a-b}{a+b} = \frac{\tan\left(\frac{A-B}{2}\right)}{\tan\left(\frac{A+B}{2}\right)} \)

22. Inscribed Angle Theorem:     \( \text{Measure of inscribed angle} = \frac{1}{2} \times \text{measure of intercepted arc} \)

23. Chord Length in a Circle:     \( c = 2r\sin\left(\frac{\theta}{2}\right) \)

C. Powers Formulas

1. Exponent Laws:  \( a^m \cdot a^n = a^{m+n} \) || \( \frac{a^m}{a^n} = a^{m-n} \) || \( (a^m)^n = a^{mn} \)

2. Negative Exponent Rule:     \( a^{-n} = \frac{1}{a^n} \)

3. Zero Exponent Rule:     \( a^0 = 1 \)

4. Power of a Quotient Rule: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)

5. Roots as Exponents: \( \sqrt[n]{a} = a^{1/n} \)

6. Product of Powers:     \( a^{mn} = (a^m)^n \)

7. Power of a Power:     \( (a^m)^n = a^{mn} \)

8. Reciprocal of a Power:     \( \frac{1}{a^n} = a^{-n} \)

9. Scientific Notation:     \( a \times 10^n \)

10. Negative Exponent Rule for Division:     \( a^{-n} = \frac{1}{a^n} \)

11. Power of a Product Rule:     \( (ab)^n = a^n \times b^n \)

12. Power of a Quotient Rule (Generalized):     \( \left(\frac{a}{b}\right)^{\frac{m}{n}} = \frac{a^{m}}{b^{n}} \)

13. Negative Exponent in the Denominator: \( \frac{1}{a^{-n}} = a^n \)

14. Zero Exponent Rule for Division: \( \frac{a^m}{a^m} = 1 \)

15. Roots as Reciprocals: \( \sqrt[n]{\frac{1}{a}} = \frac{1}{\sqrt[n]{a}} \)

16. Sum of Cubes Formula: \( a^3 + b^3 = (a + b)(a^2 – ab + b^2) \)

17. Difference of Cubes Formula: \( a^3 – b^3 = (a – b)(a^2 + ab + b^2) \)

18. Complex Conjugate Rule: \( (a + bi)(a – bi) = a^2 + b^2 \)

19. Sum of Powers Series: \( 1 + x + x^2 + \ldots + x^n = \frac{1-x^{n+1}}{1-x} \)

D. Logarithmic Formulas

1. Logarithm Properties: \( \log_b(xy) = \log_b(x) + \log_b(y) \) || \( \log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y) \)

2. Change of Base Formula: \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \)

3. Common Logarithms: \( \log(x) \text{ or } \log_{10}(x) \)

4. Natural Logarithms: \( \ln(x) \text{ or } \log_e(x) \)

5. Logarithmic Exponentiation: \( b^{\log_b(x)} = x \)

6. Logarithmic Identity: \( \log_b(b) = 1 \)

7. Logarithmic Zero: \( \log_b(1) = 0 \)

8. Logarithmic One: \( \log_b(b) = 1 \)

9. Logarithmic Inverse: \( \log_b(b^x) = x \)

10. Logarithmic Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)

11. Logarithm of One: \( \log_b(1) = 0 \)

12. Logarithm of the Base: \( \log_b(b) = 1 \)

13. Logarithm of a Power: \( \log_b(a^n) = n \cdot \log_b(a) \)

14. Exponential Form of Logarithms: \( b^{\log_b(x)} = x \)

15. Logarithm of Exponential Growth: \( \log_b(b^x) = x \)

16. Logarithmic Identity Property: \( \log_b(b) = 1 \)

17. Change of Base Formula: \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \)

18. Logarithmic Product Rule (General): \( \log_b(xy) = \frac{\log_b(x)}{\log_b(y)} \)

19. Logarithmic Quotient Rule (General): \( \log_b\left(\frac{x}{y}\right) = \frac{\log_b(x)}{\log_b(y)} \)

20. Common Logarithm Rule: \( \log(x) = \log_{10}(x) \)

E. Square Root Formulas

1. Square Root Property: \( \sqrt{a} \times \sqrt{a} = a \)

2. Product of Square Roots: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \)

3. Quotient of Square Roots: \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)

4. Square of a Sum: \( (a + b)^2 = a^2 + 2ab + b^2 \)

5. Square of a Difference: \( (a – b)^2 = a^2 – 2ab + b^2 \)

6. Conjugate Pairs: \( (a + b)(a – b) = a^2 – b^2 \)

7. Completing the Square: \( x^2 + bx + c = (x + \frac{b}{2})^2 – \left(\frac{b}{2}\right)^2 + c \)

8. Square Root of One: \( \sqrt{1} = 1 \)

9. Square Root of Zero: \( \sqrt{0} = 0 \)

10. Square Root of a Fraction: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

11. Square Root of a Decimal: \( \sqrt{0.25} = 0.5 \)

12. Square Root of a Square: \( \sqrt{a^2} = |a| \)

13. Square Root of a Cube: \( \sqrt{a^3} = a^{\frac{1}{3}} \)

14. Square Root of a Negative Number: \( \sqrt{-a} = i\sqrt{a} \)

15. Square Root of a Sum: \( \sqrt{a + b} \neq \sqrt{a} + \sqrt{b} \)

16. Square Root of Complex Numbers: \( \sqrt{a+bi} = \pm \left(\sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}} + i \sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}\right) \)

17. Square Root of Imaginary Unit: \( \sqrt{-1} = i \)

18. Square Root of a Power of 2: \( \sqrt{2^n} = 2^{n/2} \)

19. Square Root of a Rational Exponent: \( a^{m/n} = \sqrt[n]{a^m} \)

20. Square Root of a Logarithm: \( \sqrt{\log_a{x}} = a^{(\log_a{x})/2} \)