Concepts give rise to problems-1

Algebra Level 4

True or false ?

If ( c 1 a 2 − c 2 a 1 ) 2 = ( a 1 b 2 − a 2 b 1 ) ( b 1 c 2 − b 2 c 1 ) \left(c_1a_2-c_2a_1 \right)^2=\left( a_1b_2-a_2b_1\right) \left(b_1c_2-b_2c_1\right) holds true, then the equations a 1 x 2 + b 1 x + c 1 = 0 a_1x^2+b_1x +c_1=0 and a 2 x 2 + b 2 x + c 2 = 0 a_2x^2+b_2x+c_2=0 have only one common root.

False True True, if a 1 , b 1 , c 1 , a 2 , b 2 , c 2 a_1,b_1,c_1,a_2,b_2,c_2 are all real numbers True, if a 1 , b 1 , c 1 , a 2 , b 2 , c 2 a_1,b_1,c_1,a_2,b_2,c_2 are all rational numbers

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Sandeep Bhardwaj
Jul 22, 2015

Note that the above condition is satisfied if the coefficients of the two equations a 1 x 2 + b 1 x + c 1 = 0 a_1x^2+b_1x+c_1=0 and a 2 x 2 + b 2 x + c 2 = 0 a_2x^2+b_2x+c_2=0 are proportional. Hence this condition includes the situation of two common roots also.

So, if ( c 1 a 2 − c 2 a 1 ) 2 = ( a 1 b 2 − a 2 b 1 ) ( b 1 c 2 − b 2 c 1 ) \left(c_1a_2-c_2a_1 \right)^2=\left( a_1b_2-a_2b_1\right) \left(b_1c_2-b_2c_1\right) holds true, then the equations a 1 x 2 + b 1 x + c 1 = 0 a_1x^2+b_1x +c_1=0 and a 2 x 2 + b 2 x + c 2 = 0 a_2x^2+b_2x+c_2=0 have at least one common root, not necessarily exactly one common root.

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Moderator note:

RIght, the condition only implies that there is at least one root in common.

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