# ALGEBRA

**Algebra** is one of the most important topics questions from which are asked in competitive exams like **SSC, Railway** and other state level examsand play a very important role in taking edge over other students. It is estimated that 2-4 questions are always asked in SSC exams from this section in Tier 1 whereas 8-12 questions in Tier2. The difficulty level of questions depends upon your adopted approach. Some questions consumes a lot of time in calculation because of their complexity. In this article, we will learn about all algebraic formulas and their applications with shortcut tricks because all questions are based on algebraic formulas. So let’s discuss the topic in detail:

**Important Formulas: Algebra**

- 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 + 2ac + 2bc - (a + b + c)
^{3}= a^{3}+ b^{3}+ c^{3}+ 3(a + b)(b + c)(c + a) - a
^{3}+ b^{3}+ c^{3}– 3abc = (a + b+ c) (a^{2}+ b^{2}+ c^{2}– ab – ac – bc) - (a – b – c)
^{2}= a^{2}+ b^{2}+ c^{2}– 2ab – 2ac + 2bc - (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 a + b + c = 0; then a
^{3}+ b^{3}+ c^{3}=0

**Remainder theorem: **If f(x) is divided by(x – a), then the remainder is f(a).

**Factor theorem: **If (x – a) is the factor of f(x), then f(a) = 0.

**Linear Equations :**

A Linear equation is one in which each variable occurs only in its first power & not in any higher powers. Generally, we have

- One equation in one unknown
- Two equations in two unknown
- Three equations in three unknown

The general form of a system of linear equations in two variables x and y is

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x + b_{2}y + c_{2} = 0

**Set of Equations:**

**1. Consistent set of equations: **Those set of equations which have solutions.

Further these set of equations can have a unique solution if

Condition: **a ^{1} / a^{2} ≠ b^{1} / b^{2}**

These set of equations can have infinite number of solutions if

Condition: **a ^{1} / a^{2}= b^{1} / b^{2}= c^{1} / c^{2}**

**2. Inconsistent set of equations: **Those set of equations which have no solution.

Condition: **a ^{1} / a^{2} = b^{1} / b^{2 }≠ c^{1} / c^{2}**

**Quadratic Equations:**

A Quadratic equation is one in which each variable occurs only in its second power.

General form of quadratic equation is

ax^{2} + bx + c = 0

Roots are**– ****b + **** / 2a, ****– ****b – **** / 2a**

Sum of roots** = -b / a**

Product of roots **= c /a**

* *

** **

**Nature of roots**

- If b
^{2}– 4ac = 0 roots are real and equal - b
^{2}– 4ac > 0 roots are real and distinct - b
^{2}– 4ac < 0 roots are imaginary

** **

**Forming Equation from roots:**

If α and β are the roots of any quadratic equation then that equation can be written in the form

**x ^{2} – (**

**α**

**+**

**β**

**)x +**

**αβ**

**= 0**

**i.e. x ^{2} – (sum of the roots) x + Product of the roots = 0.**

** **

**Maximum /Minimum value of Quadratic Equation:**

The equation ax^{2} + bx + c = 0, will have maximum value when a < 0 and minimum value when a > 0.

The maximum or minimum values are given by

**4ac ****− ****b ^{2} / 4a**

, and will occur at x =**-b / 2a**

*Learn How to solve Algebra questions *

*Learn How to solve Algebra questions*

**Ques 1: If a ^{4} + b^{4} = a^{2}b^{2}, then (a^{6} + b^{6}) equals to?**

**Sol: **

Apply a^{3} + b^{3} = (a + b)(a^{2} – ab + b^{2}) formula;

**(a ^{6} + b^{6})** = (a

^{2})

^{3}+ (b

^{2})

^{3}

= (a^{2} + b^{2})(a^{4} + b^{4}–** **a^{2}b^{2})

= (a^{2} + b^{2}) (a^{2}b^{2}–** **a^{2}b^{2})

=0

**Question 2: **

**Solutions :**

**Question 3 : **

**Solution :**

**Question 4: **

**Solution** :

**Question 5** :

**Solutions :**

**Solutions :**

Don’t forget to share your feedback and doubts in the comment box section regarding the **Algebra tricks** discussed above which will be beneficial for your upcoming SSC exams and Railway exams preparation.

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