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in Quadratic Equations by (15 points)
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The questions x+ 2x + 3 = 0 and ax+ bx + c = 0, a, b, c belongs to R, have a common root, then a : b : c.

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Given that equations x2 + 2x + 3 = 0 and

ax2 + bx + c = 0 have a common root.

Root of x2 + 2x + 3 = 0 is 

x = \(\frac{-2\pm\sqrt{4-4\times1\times3}}2\) 

 = \(\frac{-2\pm\sqrt{4-12}}2\) 

 = \(\frac{-2\pm2\sqrt2i}2\) = -1\(\pm\)√2i

Since, roots of x2 + 2x + 3 = 0 are imaginary and it is given that one root is common with ax2 + bx + c = 0

\(\therefore\) Both roots are common of equations x2 + 2x + 3 = 0

& ax2 + bx + c = 0 (Both roots are conjugate to each other)

\(\therefore\) x2 + 2x + 3 = 0 and x2 + b/a x + c/a = 0

will represent same equation.

\(\therefore\) b/a = 2 & c/a = 3

a : b : c = a : 2a : 3a = 1 : 2 : 3.

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