Question
Factorize: $\Big(\text{x}^2+\frac{1}{\text{x}^2}\Big)-4\Big(\text{x}+\frac{1}{\text{x}}\Big)+6$
✓
Answer
$\Big(\text{x}^2+\frac{1}{\text{x}^2}\Big)-4\Big(\text{x}+\frac{1}{\text{x}}\Big)+6$
$=\text{x}^2+\frac{1}{\text{x}^2}-4\text{x}-\frac{4}{\text{x}}+4+2$
$=\text{x}^2+\frac{1}{\text{x}^2}+4+2-4\text{x}-4\text{x}$
$=\big(\text{x}^2\big)+\Big(\frac{1}{\text{x}}\Big)^2+(-2)^2+2\times\text{x}\times\frac{1}{\text{x}}+2\times\frac{1}{\text{x}}\times(-2)+2(-2)\text{x}$
Using identity $x^2 + y^2 + z^2 + 2xy + 2yz + 2zx = (x + y + z)^2$
We get, $=\Big[\text{x}+\frac{1}{\text{x}}+(-2)\Big]^2$
$=\Big[\text{x}+\frac{1}{\text{x}}-2\Big]^2$
$=\Big[\text{x}+\frac{1}{\text{x}}-2\Big]\Big[\text{x}+\frac{1}{\text{x}}-2\Big]$
$\therefore\Big[\text{x}^2+\frac{1}{\text{x}^2}\Big]-4\Big[\text{x}+\frac{1}{\text{x}}\Big]+6$
$=\Big[\text{x}+\frac{1}{\text{x}}-2\Big]\Big[\text{x}+\frac{1}{\text{x}}-2\Big]$
$=\text{x}^2+\frac{1}{\text{x}^2}-4\text{x}-\frac{4}{\text{x}}+4+2$
$=\text{x}^2+\frac{1}{\text{x}^2}+4+2-4\text{x}-4\text{x}$
$=\big(\text{x}^2\big)+\Big(\frac{1}{\text{x}}\Big)^2+(-2)^2+2\times\text{x}\times\frac{1}{\text{x}}+2\times\frac{1}{\text{x}}\times(-2)+2(-2)\text{x}$
Using identity $x^2 + y^2 + z^2 + 2xy + 2yz + 2zx = (x + y + z)^2$
We get, $=\Big[\text{x}+\frac{1}{\text{x}}+(-2)\Big]^2$
$=\Big[\text{x}+\frac{1}{\text{x}}-2\Big]^2$
$=\Big[\text{x}+\frac{1}{\text{x}}-2\Big]\Big[\text{x}+\frac{1}{\text{x}}-2\Big]$
$\therefore\Big[\text{x}^2+\frac{1}{\text{x}^2}\Big]-4\Big[\text{x}+\frac{1}{\text{x}}\Big]+6$
$=\Big[\text{x}+\frac{1}{\text{x}}-2\Big]\Big[\text{x}+\frac{1}{\text{x}}-2\Big]$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Simplify: $\frac{2\sqrt{6}}{{\sqrt2}+\sqrt{3}}+\frac{6\sqrt{2}}{{\sqrt6}+\sqrt{3}}-\frac{8\sqrt{3}}{{\sqrt6}+\sqrt{2}}$
→In a quadrilateral ABCD, bisectors of angles A and B intersect at O such that $\angle\text{AOB}=75^\circ,$ then write the value of $\angle\text{C}+\angle\text{D}.$
→In the given figure, $AB || CD || EF$. Find the value of $x$.
→→
The following table shows the daily production of $T.V.$ sets in an industry for $7$ days of a week:
Represent the above information by a pictograph.
→|
Day
|
Mon
|
Tue
|
Wed
|
Thurs
|
Fri
|
Sat
|
Sun
|
|
Number of T.V. Sets
|
$300$
|
$400$
|
$150$
|
$250$
|
$100$
|
$350$
|
$200$
|
A cylinder and a cone have equal radii of their base and equal heights. If their curved surface area are in the ratio $8 : 5$, show that the radius of each is to the height of each as $3 : 4$
→If bisectors of opposite angles of a cyclic quadrilateral $ABCD$ intersect the circle, circumscribing it at the points $P$ and $Q,$ prove that $PQ$ is a diameter of the circle.
→If $a + b + c = 9$ and $ab + bc + ca = 26$, find value of $a^3 + b^3 + c^3 - 3abc.$
→The length and breadth of a hall are in the ratio $4 : 3$ and its height is $5.5$ meters. The cost of decorating its walls (including doors and windows) at ₹ $6.60$ per square meter is ₹ $5082$. Find the length and breadth of the room.
→Two chords $AB$ and $CD$ of a circle intersect each other at $P$ outside the circle. If $AB = 6\ cm, BP = 2\ cm$ and $PD = 25\ cm$, find $CD$. 
→