Perfect Square

Algebra Level pending

For what values of k k will the expression ( k + 1 ) x 2 + 2 ( k + 3 ) x + 2 k + 3 (k+1)x^2+2(k+3)x+2k+3 be a perfect square?

We are considering perfect squares over the complex numbers. IE ( i x ) 2 = − x 2 (ix)^2 = - x^2 .

3,2 -3,-2 -3,2 3,-2

Nazmus Sakib by Nazmus sakib , 24, Bangladesh

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1 solution

Nazmus Sakib
Mar 23, 2017

for perfect square {2(k+3)}²-4.(k+1).(2k+3)=0

or,4(k²+9+6k)-4(2k²+5k+3)=0

or,-k²+6+k=0

or,k²-k-6=0

or,k²-3k+2k-6=0

or,k(k-3)+2(k-3)=0

or,(k-3)(k+2)=0

so,k=-2,3

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Technically, this is wrong because if you fill in -2, you get -x²+2x-1 = -(x-1)² which is never greater than 0 and can certainly be negative. Thus it is not a perfect square. That the discriminant is 0 just proves the equation has only one real root.

H K - 4 years, 2 months ago

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