All Necessary Algebric formula

 

1. Basic Algebraic Identities

1. $(a + b)^2 = a^2 + 2ab + b^2$
   
2. $(a - b)^2 = a^2 - 2ab + b^2$
   
3. $a^2 - b^2 = (a - b)(a + b)$
   
4. $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$
   
5. $(a - b - c)^2 = a^2 + b^2 + c^2 - 2(ab + bc - ca)$
   

2. Cubic Identities

1. $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$
   
2. $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$
   
3. $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
   
4. $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
   

3. Factorization Formulas

1. $x^2+(a+b)x+ab=(x+a)(x+b)$
2. $ax^2 + bx + c = a(x - \alpha)(x - \beta)$  ;where  $\alpha ,\mathrm{<span\; class="mspace"></span><span\; class="mord\; mathnormal">\beta \; are\; roots</span>}$

4. Quadratic Formula

For $ax^2 + bx + c = 0$:-

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

5. Logarithmic Identities

1. $\log(ab) = \log a + \log b$
   
2. $\log \left(\frac{a}{b}\right) = \log a - \log b$
   
3. $\log a^b = b \log a$
   

6. Exponential Laws

1. $a^m \cdot a^n = a^{m+n}$
   
2. $\frac{a^m}{a^n} = a^{m-n}$
   
3. $(a^m)^n = a^{mn}$
   
4. $(ab)^m = a^m b^m$
   

7. Sequence and Series

• Sum of Arithmetic Progression (AP): $$S_n = \frac{n}{2} [2a + (n-1)d]$$
  
• Sum of Geometric Progression (GP): $$S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1$$