If the equations x² + 2x + 3 = 0 and ax² + bx + c = 0 (a, b, c ∈ R) have a common root, then find the ratio a : b : — JEE Mathematics
If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0$ ($a, b, c \in \mathbb{R}$) have a common root, then find the ratio $a : b : c$.
1 Answer
VAVidaara Admin
✓ Vidaara Team
✓ Accepted
· 1mo ago
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For the equation $x^2 + 2x + 3 = 0$, the discriminant is:
$$D = 2^2 - 4(1)(3) = 4 - 12 = -8 < 0$$
Since the coefficients are real and the discriminant is negative, the roots are complex conjugates.
If a quadratic equation with real coefficients shares one complex root with another quadratic equation with real coefficients, they must share both roots.
Therefore, the two equations are identical up to a constant multiple:
$$\frac{a}{1} = \frac{b}{2} = \frac{c}{3}$$
Hence, $a : b : c = 1 : 2 : 3$.
Answer: $1 : 2 : 3$
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Discussion (5)
A
I solved it a slightly different way and got the same answer, good sign.
C
Clean and to the point. Bookmarking this for revision.
J
Clean and to the point. Bookmarking this for revision.
M
Is there a faster shortcut for this in the actual exam? Time is tight.
VA
Great discussion here. If you want more practice on this concept, check the related questions in this category.