Contents

### Algebra Formulas – What is Algebra?

**Algebra** is basically a branch of mathematics which deals with symbols and the rules for manipulating with those symbols. **Algebra** allows you to substitute the values in order to solve the equations for the unknown quantities.

### Algebra Formula

Algebra Formulas represents the relationship between the different variables. The variable can be taken as a, b, c, x, y or any other alphabet that represents a number unknown yet.

**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}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2}) - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - (a – b)
^{3}= a^{3}– 3a^{2}b + 3ab^{2}– b^{3} - (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 +…- 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^{}= 1 ;*aman*=*am*−*n*;*am*= 1*a*−*m*;*a*−*m*= 1*am*

#### **Roots of Quadratic Equation**

- For a quadratic equation ax
^{2}+ bx + c where a ≠ 0, the roots will be given by the equation as \[\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\] - Δ = b
^{2}− 4ac is called the discrimination - For real and distinct roots, Δ > 0
- For real & 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 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

#### **Factorials**

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

### Algebra Problems with a Solution

#### Solved Examples of Algebra Formulas

**Question 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 × 8

= 16**Question 2: **4

^{3}× 4

^{2}= ?

**Solution:**

Using the exponential formula (a

^{m})(a

^{n}) = a

^{m+n }where a = 4

4

^{3}× 4

^{2 }= 4

^{3+2}= 4

^{5}= 1024

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