ML Aggarwal Solution Class 9 Chapter 7 Quadratic Equations Exercise 7.1
Exercise 7.1
Question 1
Solve the following (1 to 12) equations:
(i) x2– 11x + 30 = 0
Sol :
⇒x2-11x+30=0
⇒x2-5x-6x+30=0 {∵30=−5×(−6)−11=−5−6}
⇒x(x-5)-6(x-5)=0
⇒(x-5)(x-6)=0
Either ,x-5=0, then x=5
or x-6=0, then x=6
∴x=5 , 6
(ii) 4x2-25 = 0
Sol :
⇒x2=254
∴x=±√254=±52
∴x=52,−52
Question 2
(i) 2x2-5x=0
Sol :
⇒2x2-5x=0
⇒x(2x-5)=0
Either ,x=0
or 2x-5=0, then 2x=5
⇒x=52
∴x=0,52
(ii) x2-2x=48
Sol :
⇒x2-2x-48=0
⇒x2-8x+6x-48=0 {∵−48=−8×6−2=−8+6}
⇒x(x-8)+6(x-8)=0
⇒(x-8)(x+6)=0
Either ,x-8=0, then x=8
or x+6=0, then x=-6
∴x=8, -6
Question 3
(i) 6+x=x2
Sol :
⇒6+x=x2
⇒x2-x-6=0
⇒x2-3x+2x-6=0
⇒x(x-3)+2(x-3)=0
⇒(x-3)(x+2)=0
Either ,x-3=0, then x=3
or x+2=0, then x=-2
∴x=3, 2
(ii) 2x2+3x+1=0
Sol :
⇒2x2-2x-x+1=0
⇒2x(x-1)-1(x-1)=0
⇒(x-1)(2x-1)=0
Either, x-1=0, then x=1
or 2x-1=0, then 2x=1
⇒x=12
∴x=1,12
Question 4
(i) 3x2-=2x+8
Sol :
⇒3x2=2x+8
⇒3x2-6x+4x-8=0 {∵−8×3=−24−24=−6×4−2=−6+4}
⇒3x(x-2)+4(x-2)=0
⇒(x-2)(3x+4)=0
Either, x-2=0, then x=2
or 3x+4=0,$ then 3x=-4
⇒x=−43
∴x=2,−43
(ii) 4x2+15=16x
⇒4x2-16x+15=0
⇒4x2-6x-10x+15=0 {∵4×15=60−16=−6+(−10)−16=−6−10}
⇒2x(2x-3)-5(2x-3)=0
⇒(2x-3)(2x-5)=0
Either , 2x-3=0, then 2x=3
⇒x=32
or 2x-5=0, then 2x=5
⇒x=52
∴x=32,52
Question 5
(i) x(2x+5)=25
Sol :
⇒2x2+5x-25=0
⇒2x2+10x-5x-25=0 {∵−25×2=−50−50=10×(−5)5=10−5}
⇒2x(x+5)-5(x+5)=0
⇒(x+5)(2x-5)=0
Either ,x+5=0, then x=-5
or 2x-5=0, then 2x=5
⇒x=52
∴x=−5,52
(ii) (x+3) (x–3)=40
Sol :
⇒x2-9=40
⇒x2-9-40=0
⇒x2-49=0
⇒(x)2-(7)2=0
Either , x+7=0, then x=-7
or x-7=0, then x=7
∴x=7,-7
Question 6
(i) (2x + 3) (x – 4) = 6
Sol :
⇒(2 x+3)(x-4)=6
⇒2x2-8 x+3 x-12-6=0
⇒2x2-5 x-18=0
⇒2x2-9 x+4 x-18=0 {∵−18×2=−36∴−36=−9×4−5=−9+4}
⇒x(2x-9)+2(2x-9)=0
⇒(2x-9)(x+2)=0
Either, 2x-9=0, then 2x=9
⇒x=92
or x+2=0, then x=-2
∴x=92,−2
(ii) (3x + 1) (2x + 3) = 3
Sol :
⇒6x2+9x+2x+3-3=0
⇒6x2+11x=0
⇒x(6x+11)=0
Either, x=0
or 6x+11=0, then 6x=-11
⇒x=−116
∴x=0,−116
Question 7
(i) 4x2+4x+1=0
Sol :
⇒4x2+2x+2x+1=0
⇒2x(2x+1)+1(2x+1)=0
⇒(2x+1)(2x+1)=0
Either , 2x+1=0, then x=−12
∴x=−12,−12
(ii) (x-4)2+52=132
Sol :
⇒x2-8x+16+25=169
⇒x2-8x+16+25-169=0
⇒x2-8x-128=0
⇒x2-16x+8x-128=0 {∵−128=−16×8−8=−16+8}
⇒x(x-16)+8(x-16)=0
⇒(x-16)(x+8)=0
Either ,x-16=0, then x=16
or x+8=0, then x=-8
∴x=16, 8
Question 8
(i) 21x2=4(2x+1)
Sol :
⇒21x2=4(2x+1)
⇒21x2-8x-4=0
⇒21x2-14x+6x-4=0 {∵21×(−4)=−84∴−84=−14×6−8=−14+6}
⇒7x(3x-2)+2(3x-2)=0
⇒(3x-2)(7x+2)=0
Either, 3x-2=0, then 3x=2
⇒x=23
or 7x+2=0, then 7x=-2
⇒x=−27
∴x=23,−27
(ii) 23x2−13x−1=0
Sol :
⇒2x2-x-3=0
⇒2x2-3x+2x-3=0
⇒x(2x-3)+1(2x-3)=0
⇒(2x-3)(x+1)=0
Either,2x-3=0, then 2x=3
⇒x=32
or x+1=0, then x=-1
∴x=32,−1
Question 9
(i) 6x+29=5x
Sol :
⇒6x2+29x-5=0
⇒6x2+30x-x-5=0 {∵6×(−5)=−30∴−30=30×(−1)29=30−1}
⇒6x(x+5)-1(x+5)=0
⇒(x+5)(6x-1)=0
Either,x=+5=0, then x=-5
or 6x-1=0, then 6x=1
⇒x=16
∴x=15,−5
(ii) x+1x=212
Sol :
⇒x+1x=52
⇒x2+1=52x
⇒x2−52x+1=0
⇒2x2-5x+2=0
⇒2x2-x-4x+2=0 {∵2×2=44=−1×(−4)−5=−1−4}
⇒x(2x-1)-2(2x-1)=0
⇒(2x-1)(x-2)=0
Either ,2x-1=0, then 2x=1
⇒x=12 or x-2=0, then x=2
∴x=2,12
Question 10
(i) 3x−8x=2
Sol :
⇒3x2−8=2xx
⇒3x2-2x-8=0
⇒3x2-6x+4x-8=0 {∵−8×3=−24−24=−6×4−2=−6+4}
⇒3x(x-2)+4(x-2)=0
⇒(x-2)(3x+4)=0
Either, x-2=0, then x=2
or 3x+4=0, then 3x=-4
⇒x=−43
∴x=2,−43
(ii) x3+9x=4
Sol :
⇒x2+27=12x3x
⇒x2-12x+27=0
⇒x2-3x-9x+27=0 {∵27=−3×(−9)−12=−3−9}
⇒x(x-3)-9(x-3)=0
⇒(x-3)(x-9)=0
Either ,x-3=0, then x=3
or x-9=0, then x=9
Question 11
(i) x−1x+1=2x−53x−7
Sol :
⇒x−1x+1=2x−53x−7
By cross multiplication
⇒(x-1)(3x-7)=(x+1)(2x-5)
⇒3x2-7x-3x+7=2x2-5x+2x-5
⇒3x2-10x+7=2x2-3x-5=0
⇒3x2-10x+7-2x2+3x+5=0
⇒x2-7x+12=0
⇒x2-4x-3x+12=0
⇒x(x-4)-3(x-4)=0 {∵12=−4×(−3)−7=−4−3}
⇒(x-4)(x-3)=0
Either ,x--4=0, then x=4
or x-3=0, then x=3
∴x=3 ,4
(ii) 1x+2+1x=34
Sol :
⇒x+x+2x(x+2)=34
⇒2x+2x(x+2)=34
by cross multiplication
⇒3x(x+2)=4(2x+2)
⇒3x2+6x=8x+8
⇒3x2+6x-8x-8=0
⇒3x2-2x-8=0
⇒3x2-6x+4x-8=0 {∵3×(−8)=−24∴−24=−6×4−2=−6+4}
⇒3x(x-2)+4(x-2)=0
⇒(x-2)(3x+4)=0
Either , x-2=0, then x=2
or 3x+4=0, then 3x=-4
⇒x=−43
Hence , x=2,−43
Question 12
(i) 8x+3−32−x=2
Sol :
⇒8(2−x)−3(x+3)(x+3)(2−x)=21
⇒16−8x−3x−92x−x2+6−3x=21
⇒7−11x−x2−x+6=21
⇒7-11x=-2x2-2x+12 (by cross multiplication )
⇒2x2+2x-12+7-11x=0
⇒2x2-9x-5=0
⇒2x2-10x+x-5=0 {∵2×(−5)=−10∴−10=−10×1−9=−10+1}
⇒2x(x-5)+1(x-5)=0
⇒(x-5)(2x+1)=0
Either , x-5=0, then x=5
or 2x+1=0, then 2x=-1
⇒x=−12
∴x=5,−12
(ii) xx+1+x+1x=216
Sol :
⇒x2+(x+1)2x(x+1)=136
⇒x2+x2+2x+1x2+x=136
⇒2x2+2x+1x2+x=136
⇒13x2+13x=12x2+12x+6
⇒13x2+13x-12xx2-12x-6=0
⇒x2+x-6=0
⇒x2+3x-2x-6=0 {∵−6=3×(−2)1=3−2}
⇒x(x+3)-2(x+3)=0
⇒(x+3)(x-0)=0
⇒x(x+3)-2(x+3)=0
⇒(x+3)(x-0)=0
Either ,x+3=0, then x=-3
or x-2=0, then x=2
∴x=2,3
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