Algebra is a branch of mathematics that substitutes letters for numbers. An algebraic equation depicts a scale, what is done on one side of the scale with a number is also done to either side of the scale. The numbers are constants. Algebra also includes real numbers, complex numbers, matrices, vectors and much more. X, Y, A, B are the most commonly used letters that represent algebraic problems and equations.

Algebra Formulas from Class 8 to Class 12 | Algebra Formulas For Class 8 | Algebra Formulas For Class 9 | Algebra Formulas For Class 10 | Algebra Formulas For Class 11 | Algebra Formulas For Class 12 |
---|

## Important Formulas in Algebra

**Here is a list of Algebraic formulas** –

- a
^{2}– b^{2}= (a – b)(a + b) - (a + b)
^{2}= a^{2}+ 2ab + b^{2} - a
^{2}+ b^{2}= (a + b)^{2}– 2ab - (a – b)
^{2}= a^{2}– 2ab + b^{2} - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ca - (a – b – c)
^{2}= a^{2}+ b^{2}+ c^{2}– 2ab + 2bc – 2ca - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}; (a + b)^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a – b)
^{3}= a^{3}– 3a^{2}b + 3ab^{2}– b^{3 }= a^{3}– b^{3}– 3ab(a – b) - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2}) - (a + b)
^{4}= a^{4}+ 4a^{3}b + 6a^{2}b^{2}+ 4ab^{3}+ b^{4} - (a – b)
^{4}= a^{4}– 4a^{3}b + 6a^{2}b^{2}– 4ab^{3}+ b^{4} - a
^{4}– b^{4}= (a – b)(a + b)(a^{2}+ b^{2}) - a
^{5}– b^{5}= (a – b)(a^{4}+ a^{3}b + a^{2}b^{2}+ ab^{3}+ b^{4}) **If n is a natural number**a^{n}– b^{n}= (a – b)(a^{n-1}+ a^{n-2}b+…+ b^{n-2}a + b^{n-1})**If n is even**(n = 2k), a^{n}+ b^{n}= (a + b)(a^{n-1}– a^{n-2}b +…+ b^{n-2}a – b^{n-1})**If n is odd**(n = 2k + 1), a^{n}+ b^{n}= (a + b)(a^{n-1}– a^{n-2}b +a^{n-3}b^{2}…- b^{n-2}a + b^{n-1})- (a + b + c + …)
^{2}= a^{2}+ b^{2}+ c^{2}+ … + 2(ab + ac + bc + ….) **Laws of Exponents**(a^{m})(a^{n}) = a^{m+n}; (ab)^{m}= a^{m}b^{m }; (a^{m})^{n}= a^{mn}**Fractional Exponents**a^{0}= 1 ; $\frac{a^{m}}{a^{n}} = a^{m-n}$ ; $a^{m}$ = $\frac{1}{a^{-m}}$ ; $a^{-m}$ = $\frac{1}{a^{m}}$

**Roots of Quadratic Equation**

- For a quadratic equation ax
^{2}+ bx + c = 0 where a ≠ 0, the roots will be given by the equation as \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\) - Δ = b
^{2}− 4ac is called the discriminant - For real and distinct roots, Δ > 0
- For real and coincident roots, Δ = 0
- For non-real roots, Δ < 0
- If α and β are the two roots of the equation ax
^{2}+ bx + c = 0 then, α + β = (-b / a) and α × β = (c / a). - If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0

- For a quadratic equation ax

**Factorials**

- n! = (1).(2).(3)…..(n − 1).n
- n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
- 0! = 1
- \((a + b)^{n} = a^{n}+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^{2}+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^{3}+….+b^{n}, where\;,n>1\)

### Solved Examples

**Example 1:**Find out the value of 5

^{2}– 3

^{2 }

**Solution:**

Using the formula a

^{2}– b

^{2}= (a – b)(a + b)

where a = 5 and b = 3

(a – b)(a + b)

= (5 – 3)(5 + 3)

= 2 $\times$ 8

= 16

**4**

Example 2:

Example 2:

^{3}$\times$ 4

^{2}= ?

**Solution:**

Using the exponential formula (a

^{m})(a

^{n}) = a

^{m+n }where a = 4

4

^{3}$\times$ 4

^{2 }= 4

^{3+2 }= 4

^{5 }= 1024