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Algebra Level 3

The equation ( 1 + n 2 ) x 2 + 2 n c x + ( c 2 − a 2 ) = 0 (1+n^2)x^2+2ncx+(c^2-a^2)=0 will have equal roots if

c^2=1+a^2 c^2=(1+n^2)a^2 c^2=1+n^2+a^2 c^2=1-a^2

Aditya Khotte by Aditya Khotte , 21, India

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1 solution

Hung Woei Neoh
Apr 22, 2016

For equal roots, the discriminant b 2 − 4 a c = 0 b^2-4ac =0

( 2 n c ) 2 − 4 ( 1 + n 2 ) ( c 2 − a 2 ) = 0 4 n 2 c 2 − 4 c 2 + 4 a 2 − 4 n 2 c 2 + 4 n 2 a 2 = 0 4 c 2 = 4 a 2 + 4 n 2 a 2 c 2 = a 2 + n 2 a 2 c 2 = ( 1 + n 2 ) a 2 (2nc)^2 - 4(1+n^2)(c^2-a^2) = 0\\ 4n^2c^2 - 4c^2 + 4a^2 - 4n^2c^2 +4n^2a^2 = 0\\ 4c^2 = 4a^2 + 4n^2a^2\\ c^2 = a^2 + n^2a^2\\ \boxed{c^2 = (1+n^2)a^2}

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