In the following question find the ratio of a:b:cIf the equations x2+2x+3=0 and ax2+bx+c=0; a,b,c∈R, have a common root, then a:b:c is ?

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In the following question find the ratio of a:b:cIf the equations x2+2x+3=0 and ax2+bx+c=0; a,b,c∈R, have a common root, then a:b:c is ?

Eliza Flores · Accepted Answer

Here is a direct answer to your question. You should have continued with a Gauss decomposition of the quadratic expression:9a2+3b2+c2-6ab-2ac-2bc=0&amp;nbsp;&amp;nbsp;&amp;nbsp;(1)&amp;nbsp;into:(3a−b−c/3)2+2(b−2c/3)2=0&amp;nbsp;&amp;nbsp;&amp;nbsp;⟺&amp;nbsp;&amp;nbsp;&amp;nbsp;{3a−b−c/3=0Eq.(2)b−2c/3=0Eq.(3)From&amp;nbsp;(3), one deduces that&amp;nbsp;b=2c3; plugging this expression of b into&amp;nbsp;(2)&amp;nbsp;gives&amp;nbsp;3a=c. Therefore we have established the desired proportionality.Remark 1: expression&amp;nbsp;(1)&amp;nbsp;can be directly obtained by calculating the so-called resultantdet(12300123abc00abc)Remark 2: There was a completely different way to solve this question. As the roots of the first polynomial are&amp;nbsp;−1+iξ3, plugging (one of) them into the second equation gives:(−2a−b+c)+iε3(−2a+b)=0Identifiying real part and imaginary part with zero gives at once the result.

Yaretzi Odom · Accepted Answer

Discriminant is negative. It means the roots are complex conjugate of each other. So, both equations have both roots common. So, a:b:c=1:2:3.

In the following question find the ratio of a:b:c If the

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