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Question

Solve the following equation:
12x2+1x2-56x+1x+89=0

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Solution

We have 12x2+1x2 -56x+1x+89=0We know the identity x2+1x2=x+1x2-2Thus, the equation can be written as: 12x+1x2-2-56x+1x+89=012x+1x2-24-56x+1x+89=012x+1x2-56x+1x+65=0Let m=x+1x. Then, the equation can be further written as:12m2-56m+65=0On using the quadratic formula, we get: m=--56±-562-41265212 =56±3136-312024 =56±1624 =56±424 = 6024, 5224 =52,136On substituting m=x+1x, we get:x+1x=52....(1) or x+1x=136...(2)

Now from equation (1), we get: x+1x=52x2+1x=522x2+2=5x2x2-5x+2=0On splitting the middle term -5x as -x-4x, we get: 2x2-x-4x+2=0x2x-1-22x-1=02x-1x-2=02x-1=0 or x-2=0x=12 or x=2Now, from equation (2), we get: x+1x=136x2+1x=1366x2+6=13x6x2-13x+6=0

On splitting the middle term -13x as -9x-4x, we get: 6x2-9x-4x+6=03x2x-3-22x-3=02x-33x-2=02x-3=0 or 3x-2=0x=32 or x=23Thus, the solutions of the given equation are x=12,2,23,32.

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