If a and b are the zeros of polynomial kx2+ 4 x + 4 show that

a2+ b2 = 24 and also find value of k.

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Solution

a,b are the zeros of f(x)=kx²+4x+4 and a²+b²=24
a+b=-4/k and a×b=4/k
now, a²+b²=24
or, (a+b)²-2ab=24
or, (-4/k)²-2×4/k=24
or, 16/k²-8/k=24
or, (16-8k)/k²=24
or, 16-8k=24k²
or, 24k²+8k-16=0
or, 8(3k²+k-2)=0
or, 3k²+k-2=0
or, 3k²+3k-2k-2=0
or, 3k(k+1)-2(k+1)=0
or, (k+1)(3k-2)=0
either, k+1=0; or, k=-1
or, 3k-2=0; or, k=2/3

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