Question
Factories: $\left(a^2-54\right)^2-36$
✓
Answer
$\left(a^2-54\right)^2-36=\left(a^2-5 a\right)^2-6^2$
$=\left[\left(a^2-5 a\right)-6\right]\left[\left(a^2-5 a\right)+6\right]$
$=\left(a^2-5 a-6\right)\left(a^2-5 a+6\right)$
In order to factories $a^2-5 a-6$, we will find two number $p$ and $q$ such that $p+q=-5$ and $p q=-6$
Now,
$(-6)+1=-5 \text { And }(-6) \times 1=-6$
Splitting the middle term -5 in the given quadratic as $-6+a$, we get:
$a^2-5 a-6=a^2-6 a+a-6$
$=\left(a^2-6 a\right)+(a-6)$
$=a(a-6)+(a-6)$
$=(a+1)(a-6)$
Now, In order to factories $a^2-5 a+6$, we will find two numbers $p$ and $q$ such that $p+q=-5$ and $p q=6$ Clearly,
$(-2)+(-3)=-5 \text { and }(-2) \times(-3)=6$
Splitting the middle term -5 in the quadratic as $-2 a-3 a$, we get:
$a^2-5 a+6=a^2-2 a-3 a+6$
$=\left(a^2-2 a\right)-(3 a-6)$
$=a(a-2)-3(a-2)$
$=(a-3)(a-2)$
$\therefore\left(a^2-5 a-6\right)\left(a^2-5 a+6\right)$
$=(a-6)(a+1)(a-3)(a-2)$
$=(a+1)(a-2)(a-3)(a-6)$
$=\left[\left(a^2-5 a\right)-6\right]\left[\left(a^2-5 a\right)+6\right]$
$=\left(a^2-5 a-6\right)\left(a^2-5 a+6\right)$
In order to factories $a^2-5 a-6$, we will find two number $p$ and $q$ such that $p+q=-5$ and $p q=-6$
Now,
$(-6)+1=-5 \text { And }(-6) \times 1=-6$
Splitting the middle term -5 in the given quadratic as $-6+a$, we get:
$a^2-5 a-6=a^2-6 a+a-6$
$=\left(a^2-6 a\right)+(a-6)$
$=a(a-6)+(a-6)$
$=(a+1)(a-6)$
Now, In order to factories $a^2-5 a+6$, we will find two numbers $p$ and $q$ such that $p+q=-5$ and $p q=6$ Clearly,
$(-2)+(-3)=-5 \text { and }(-2) \times(-3)=6$
Splitting the middle term -5 in the quadratic as $-2 a-3 a$, we get:
$a^2-5 a+6=a^2-2 a-3 a+6$
$=\left(a^2-2 a\right)-(3 a-6)$
$=a(a-2)-3(a-2)$
$=(a-3)(a-2)$
$\therefore\left(a^2-5 a-6\right)\left(a^2-5 a+6\right)$
$=(a-6)(a+1)(a-3)(a-2)$
$=(a+1)(a-2)(a-3)(a-6)$
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
Construct a rectangle PQRS in which QR = 3.6cm and diagonal PR = 6cm. Measure the other side of the rectangle.
Solve the following equation and verify your answer:
$\frac{3\text{x}+5}{2\text{x}+7}=4$
→$\frac{3\text{x}+5}{2\text{x}+7}=4$
Simplify :
(3r – 2k)³ + (3r + 2k)³
The top speeds of 30 different land animals have been organised into a frequency table given below:
Draw a histogram for the given data.
|
Maximum speed (in km/ h)
|
10-20
|
20-30
|
30-40
|
40-50
|
50-60
|
60-70
|
|
Number of animals
|
3
|
5
|
10
|
8
|
0
|
2
|
Find the areas of given plots. (All measures are in meters.)
The following table shows the expenditure incurred by a publisher in publishing a book:
Present the above data in the form of a pie-chart.
|
Items
|
Paper
|
Printing
|
Binding
|
Advertising
|
Miscellaneous
|
|
Expenditure (in %)
|
35%
|
20%
|
10%
|
5%
|
30%
|
Leonard Euler, discovered an interesting formula regarding the faces, vertices and edges of solid figures.
Count and write the faces, vertices and edges of the following figures and complete the table. From the table verify Euler’s formula, F + V = E + 2.
Count and write the faces, vertices and edges of the following figures and complete the table. From the table verify Euler’s formula, F + V = E + 2.
Making use of the cube root table, find the cube root 7532
Construct, if possible, a quadrilateral ABCD given AB = 6cm, BC = 3.7cm, CD = 5.7cm, AD = 5.5cm and BD = 6.1cm. Give reasons for not being able to construct it, if you cannot.
The marks obtained by Sudhir in an examination are given below:
Represent the above data by a pie chart.
|
Type of fruit
|
Mangoes
|
Apples
|
Oranges
|
Coconuts
|
Pomegranates
|
|
Number
|
26
|
30
|
21
|
5
|
8
|