Cracku | Algebra Cheatsheet

Theory

Algebra is a very important topic for SSC exams. Every year, it is expected that 2-4 questions can be asked on this topic. The difficulty level of the question will depend upon your approach to solve the question. Most of the questions can be solved easily using formulas and shortcuts while a few of them can be complex and takes a lot of time.

Formula

$$(a+b)^2 = a^2+b^2+2ab$$
$$(a-b)^2 = a^2+b^2-2ab$$
$$(a+b)^2+(a+b)^2 = 2a^2+2b^2$$
$$(a+b)^2-(a-b)^2 = 4ab$$
$$a^3+b^3 = (a+b)(a^2-ab+b^2)$$
$$a^3-b^3 = (a-b)(a^2+ab+b^2)$$
$$a^2-b^2 = (a+b)(a-b)$$
$$a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
If a+b+c = 0, then $$a^3+b^3+c^3 = 3abc$$
$$(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$$
$$(a+b)^3 = a^3+b^3+3ab(a+b)$$
$$(a-b)^3 = a^3+b^3-3ab(a-b)$$

Shortcuts

If $$x+\frac{1}{x}=a$$, then $$x^2+\frac{1}{x^2} = a^2-2$$
If $$x+\frac{1}{x}=a$$, then $$x^4+\frac{1}{x^4} = (a^2-2)^2-2 = a^4-4a^2+2$$
If $$x-\frac{1}{x}=a$$, then $$x^2+\frac{1}{x^2} = a^2+2$$
If $$x-\frac{1}{x}=a$$, then $$x^4+\frac{1}{x^4} = (a^2+2)^2-2 = a^4+4a^2+2$$
If $$x^4+\frac{1}{x^4} = a$$, then $$x^2+\frac{1}{x^2} = \sqrt{a+2}$$
If $$x^4+\frac{1}{x^4} = a$$, then $$x+\frac{1}{x} = \sqrt{\sqrt{a+2}+2}$$
If $$x^4+\frac{1}{x^4} = a$$, then $$x-\frac{1}{x} = \sqrt{\sqrt{a+2}-2}$$
If $$x+\dfrac{1}{x}=1$$, then $$x^3=-1$$
If $$x+\dfrac{1}{x}=-1$$, then $$x^3=1$$

Solved Example

Q) If $$x-\frac{1}{x} = 3$$, then find the value of $$x^4+\frac{1}{x^4}$$.

Solution:
Given, $$x-\frac{1}{x} = 3$$
$$x^4+\frac{1}{x^4} = a^4+4a^2+2 = 3^4+4\times3^2+2 = 81+36+2 = 119$$

Solved Example

Q) If $$x^4+\dfrac{1}{x^4} = 119$$, then find the value of $$x+\dfrac{1}{x}$$

Solution:
Given, $$x^4+\dfrac{1}{x^4} = 119$$
$$x+\dfrac{1}{x} = \sqrt{\sqrt{119+2}+2} = \sqrt{\sqrt{121}+2} = \sqrt{11+2} = \sqrt{13}$$

Algebra

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