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Question

Solve the following equations.

(i) x2 + 12x -203x - 5 = x2 + 8x + 122x + 3

(ii) 10x2 + 15x + 635x2 - 25x + 12= 2x + 3x - 5

(iii) 2x + 12 + 2x - 122x + 12 - 2x - 12= 178

(iv) 4x + 1 + x + 34x + 1 -x + 3= 41

(v) 4x + 1 2 + 2x + 324x2 + 12x + 9= 6136

(vi) 3x - 4 3 - x + 1 33x - 4 3 + x + 1 3 = 61189

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Solution


(i)
x2+12x-203x-5=x2+8x+122x+3

Multiplying both sides by 14, we get

x2+12x-2012x-20=x2+8x+128x+12

Using dividendo, we get

x2+12x-20-12x-2012x-20=x2+8x+12-8x+128x+12x2+12x-20-12x+2012x-20=x2+8x+12-8x-128x+12x212x-20=x28x+12
This equation is true for x = 0. Therefore, x = 0 is a solution of the given equation.

If x ≠ 0, then x2 ≠ 0.

Dividing both sides by x2, we get

112x-20=18x+1212x-20=8x+1212x-8x=20+124x=32x=8
Thus, the solutions of the given equation are x = 0 and x = 8.

(ii)
10x2+15x+635x2-25x+12=2x+3x-5 10x2+15x+632x+3=5x2-25x+12x-5
Multiplying both sides by 15x, we get

10x2+15x+6310x2+15x=5x2-25x+125x2-25x

Using dividendo, we get

10x2+15x+63-10x2+15x10x2+15x=5x2-25x+12-5x2-25x5x2-25x6310x2+15x=125x2-25x635x2x+3=125xx-5632x+3=12x-5
63x-315=24x+3663x-24x=315+3639x=351x=35139=9
Thus, the solution of the give equation is x = 9.

(iii)
2x+12 +2x-122x+12 -2x-12=178

Applying componendo and dividendo, we get

2x+12 +2x-12+2x+12 -2x-122x+12 +2x-12-2x+12 -2x-12=17+817-822x+12 22x-12=2592x+12x-1=259=53
6x+3=10x-510x-6x=5+34x=8x=2
Thus, the solution of the given equation is x = 2.

(iv)
4x+1+x+34x+1-x+3=41

Applying componendo and dividendo, we get

4x+1+x+3+4x+1-x+34x+1+x+3-4x+1-x+3=4+14-124x+12x+3=534x+1x+3=53
Squaring on both sides, we get

4x+1x+3=532=25936x+9=25x+7536x-25x=75-911x=66x=6
Thus, the solution of the given equation is x = 6.

(v)
4x+12 +2x+324x2+12x+9=61364x+12+2x+322x+32=6136
Applying dividendo, we get

4x+12+2x+32-2x+322x+32=61-36364x+122x+32=2536
Taking square root on both sides, we get

4x+12x+3=2536=5624x+6=10x+1524x-10x=15-614x=9x=914
Thus, the solution of the given equation is x=914.

(vi)
3x-43-x+133x-43+x+13=61189

Applying componendo and dividendo, we get

3x-43-x+13+3x-43+x+133x-43-x+13-3x-43+x+13=61+18961-18923x-43-2x+13=250-1283x-43x+13=12564
Taking cube root on both sides, we get

3x-4x+1=1256433x-4x+1=5433=5412x-16=5x+512x-5x=16+57x=21x=3
Thus, the solution of the given equation is x = 3.

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