Factorisation Class 9 RS Aggarwal Exe-4B Goyal Brothers ICSE Maths Solutions Ch-4. Expansions Class 9 RS Aggarwal Exe-3A Goyal Brothers ICSE Foundation Maths Solutions. In this article you will learn Differences of two Squire Using Formula easily. Visit official Website CISCE for detail information about ICSE Board Class-9 Maths.
Factorisation Class 9 RS Aggarwal Exe-4B Goyal Brothers ICSE Maths Solutions Ch-4
| Board | ICSE |
| Subject | Maths |
| Class | 9th |
| Chapter-4 | Factorisation |
| Writer | RS Aggrawal |
| Topics | Differences of two Squire Using Formula |
| Academic Session | 2024-2025 |
Differences of two Squire Using Formula
When an expression is as the difference of two perfect squares, i.e. a²- b², then we can factor it as (a+b) (a-b).
Exercise- 4B
Factorise (Q1 to 30) : Page- 61
Que-1: x²-49
Sol: (x)²-(7)²
(x+7)(x-7).
Que-2: 25x²-64y²
Sol: 25x²-64y²
(5x)² – (8y)²
(5x-8y)(5x+8y).
Que-3: 100-9p²
Sol: 100-9p²
(10)²-(3p)²
(10-3p)(10+3p).
Que-4: 80-5a²
Sol: 80-5a²
5(16-a²)
5[(4)²-(a)²]
5(4-a)(4+a).
Que-5: 32x²-18y²
Sol: 32x²-18y²
2[16x²-9y²]
2[(4x)²-(3y)²]
2(4x-3y)(4x+3y).
Que-6: 3x³-48x
Sol: 3x³-48x
3x(x²-16)
3x[(x)²-(4)²]
3x(x-4)(x+4).
Que-7: x^4 – 81
Sol: x^4 – 81
(x²)²-(9)²
(x²-9)(x²+9)
[(x)²-(3)²] (x²+9)
(x²+9)(x-3)(x+3).
Que-8: 2x^4 – 32
Sol: 2x^4 – 32
2(x^4-16)
2[(x²)²-(4)²]
2(x²+4)(x²-4)
2(x²+4)[(x)²-(2)²]
2(x²+4)(x-2)(x+2).
Que-9: x³-5x²-x+5
Sol: x³-5x²-x+5
x²(x-5)-1(x-5)
(x-5)(x²-1)
(x-5)[(x)²-(1)²]
(x-5)(x-1)(x+1).
Que-10: 9(x+a)²-4x²
Sol: 9(x+a)²-4x²
[3(x+a)]² – (2x)²
[3(x+a)+2x][3(x+a)-2x]
[3x+3a+2x][3x+3a-2x]
[5x+3a][x+3a].
Que-11: 9(b+2a)²-4a²
Sol: 9(b+2a)²-4a²
[3(b+2a)]² – (2a)²
[3(b+2a)-2a][3(b+2a)+2a]
[3b+6a-2a][3b+6a+2a]
[3b+4a][3b+8a].
Que-12: 3-12(a-b)²
Sol: 3-12(a-b)²
3[1-4(a-b)²]
3[(1)²-{2(a-b)}²]
3[1-{2(a-b)}][1+{2(a-b)}]
3[1-2a+2b][1+2a-2b].
Que-13: 50a²-2(b-c)²
Sol: 50a²-2(b-c)²
2[25a²-(b-c)²]
2[(5a)²-(b-c)²]
2[5a-(b-c)][5a+(b-c)]
2[5a-b+c][5a+b-c]
Que-14: 2(x-3)²-32
Sol: 2(x-3)²-32
2[(x-3)²-16]
2[(x-3)-(4)²]
2[(x-3)-4][(x-3)+4]
2(x-7)(x+1).
Que-15: a²(b+c)-(b+c)²
Sol: a²(b+c)-(b+c)²
(b+c)[a²-(b+c)²]
(b+c)[a-(b+c)][a+(b+c)]
(b+c)(a-b-c)(a+b+c).
Que-16: x²-1-2a-a²
Sol: x²-1-2a-a²
x²-(a²+2a+1)
x²-(a+1)²
(x-a-1)(x+a+1).
Que-17: x²-y²+2yz-z²
Sol: x²-y²+2yz-z²
x²-(y²-2yz+z²)
x²-(y-z)²
(x-y+z)(x+y-z).
Que-18: x²-y²-4xz+4z²
Sol: x²-y²-4xz+4z²
[x²-4xz+(2z)²]-(y)²
(x-2z)²-(y)²
(x-2z-y)(x-2z+y).
Que-19: x²-4x+4y-y²
Sol: x²-4x+4y-y²
x²-y²-4x+4y
(x)²-(y)²-4(x-y)
(x+y)(x-y)-4(x-y)
(x-y)(x+y-4)
Que-20: x-y-x²+y²
Sol: x-y-x²+y²
(x-y)-(x²-y²)
(x-y)-[(x-y)(x+y)]
(x-y)(1-x-y).
Que-21: x(x+z)-y(y+z)
Sol: x(x+z)-y(y+z)
x²+xz-y²-yz
x²-y²+xz-yz
(x)²-(y)²+z(x-y)
(x+y)(x-y)+z(x-y)
(x-y)(x+y+z).
Que-22: x(x-2)-y(y-2)
Sol: x(x-2)-y(y-2)
x²-2x-y²+2y
x²-y²-2x+2y
(x)²-(y)²-2(x-y)
(x-y)(x+y)-2(x-y)
(x-y)(x+y-2).
Que-23: 4x²y-9y³
Sol: 4x²y-9y³
y(4x²-9y²)
y[(2x)²-(3y)²]
y(2x-3y)(2x+3y).
Que-24: 9x^4 – x²-12x-36
Sol: 9x^4 – x²-12x-36
9x^4 – [x²+12x+36]
(3x²)² – [(x)²+12x+(6)²]
(3x²)² – (x+6)²
(3x²-x-6)(3x²+x+6)
Que-25: x²+(1/x²)-11
Sol: x²+(1/x²)-11
[x²+(1/x²)-2]-9
[x-(1/x)]² – (3)²
[x-(1/x)+3][x-(1/x)-3].
Que-26: x^4 + 5x²+9
Sol: x^4 + 5x²+9
x^4 +9+6x²-x²
[(x²)²+(3)²+6x²] – (x)²
(x²+3)² – (x)²
(x²+3-x)(x²+3+x).
Que-27: a²+b²-c²-d²+2ab-2cd
Sol: a²+b²−c²−d²+2ab−2cd
rearrange the terms ,
= a²+b²+2ab−c²−d²−2cd
= (a²+b²+2ab)−(c²+d²+2cd)
= (a+b)²−(c+d)²
= [(a+b)+(c+d)] [(a+b)−(c+d)]
= (a+b+c+d) (a+b−c−d)
Que-28: (a²-b²)(c²-d²)-4abcd
Sol: (a²-b²)(c²-d²) – 4abcd = a²c² – a²d² – b²c² + b²d² – 2abcd – 2abcd
= a²c² + b²d² – 2abcd – ( a²d² + b²c² + 2abcd ) = (ac-bd)² – (ad+ bc)²
[using the formula p² – q² = (p-q)(p+q) ]
= (ac-bd – ad – bc) (ac -bd + ad +bc).
Que-29: 4x²-12ax-y²-z²-2yz+9a²
Sol: 4x²−12ax−y²−z²−2yz+9a²
= (4x²+9a²−12ax)−(y²+z²+2yz)
= (2x−3a)²−(y+z)²
= [(2x−3a)+(y+z)] [(2x−3a)−(y+z)]
= (2x−3a+y+z)(2x−3a−y−z).
Que-30: 9a²+3a-8b-64b²
Sol: 9a²+3a-8b-64b²
9a²-64b²+3a-8b
[(3a)²-(8b)²] + (3a-8b)
(3a+8b)(3a-8b)+1(3a-8b)
(3a+8b+1)(3a-8b).
Que-31: Express (x²+8x-15)(x²-8x-15) as the difference of two squares.
Sol: (x²+8x−15)(x²−8x−15)
= [(x²−15)+(8x)] [(x²−15)−(8x)]
=(x²−15)²−(8x)² ….[Using a²−b²=(a+b)(a−b)]
Que-32: Evaluate:
(i) [(674)² – (326)²] (ii) [(18.6)² – (1.4)²]
Sol: [(674)²-(326)²]
[Using a²−b²=(a+b)(a−b)]
(674-326)(674+326)
348 x 1000
= 348000.
(ii) [(18.6)²-(1.4)²]
[Using a²−b²=(a+b)(a−b)]
(18.6+1.4)(18.6-1.4)
20 x 17.2
= 344.0
Que-33: Factorise: (x^4 + x²y² +y^4)
Sol: x^4+x²y²+y^4
= (x²)²+2x²y²+y^4−x²y²
= (x²+y²)²−(xy)² {∵ a²−b² = (a+b)(a−b)}
= (x²+y²+xy)(x²+y²−xy).
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