If the equations x² + 2x + 3 = 0 and ax² + bx + c = 0, a, b, c ∈ R, have a common root, then a : b :

Q: Wait, why do we reject the negative root here? Or why can't we just use energy conservation?

This is a common point of confusion! In this problem, we aren't dealing with energy conservation, but rather the fundamental properties of polynomials. We don't 'reject' a root; rather, we discover that the roots are complex. Because the coefficients are real, complex roots must come in pairs. If one is a root, its conjugate must also be a root. This forces both roots to be shared between the two equations.

Q: What if the coefficients $a, b, c$ were not real? Would the answer change?

Absolutely! The Conjugate Pair Theorem relies entirely on the coefficients being real numbers. If $a, b, c$ were complex, we could not guarantee that the conjugate of a root is also a root. In that case, the equations might only share one root, and the proportionality condition would not necessarily hold.

Q: How do I know for sure that *both* roots are common?

Great question. We know the first equation $x^2 + 2x + 3 = 0$ has complex roots. Let's call them $\alpha$ and $\bar{\alpha}$. Since the second equation $ax^2 + bx + c = 0$ has real coefficients and shares the root $\alpha$, it must also share the root $\bar{\alpha}$ by the Conjugate Pair Theorem. Thus, both roots are shared, which forces the equations to be proportional.

Question

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0, a, b, c \in R$, have a common root, then $a : b : c$ is

Options:

(A) 1 : 2 : 3

(B) 3 : 2 : 1

(C) 1 : 3 : 2

(D) 3 : 1 : 2

Sigma Percentile · Accepted Answer

### Analyzing the Parabola

Imagine you are standing before the graph of the quadratic equation $x^2 + 2x + 3 = 0$. If you were to plot this as a function $y = x^2 + 2x + 3$, you would see a beautiful, upward-opening parabola.

To determine its position relative to the $x$-axis, we utilize the discriminant:

$$D = B^2 - 4AC$$

By substituting our coefficients—$A=1$, $B=2$, $C=3$—we find:

$$D = (2)^2 - 4(1)(3) = 4 - 12 = -8$$

Because $D < 0$, the parabola never touches the $x$-axis; it floats entirely above it. This confirms that the roots are not real, but rather **complex**.

### The Power of the Conjugate Pair

Now, consider a second equation $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$. We are told it shares a common root with our first equation.

Because the first equation has complex roots and the second equation has real coefficients, the **Conjugate Pair Theorem** applies. This theorem states that if a polynomial with real coefficients has a complex root $p + iq$, it *must* also have the conjugate $p - iq$ as a root.

It is a "package deal." Therefore, if the second equation shares one complex root, it is mathematically forced to share the other as well. **Both roots are common.**

### The Elegance of Proportionality

Once we realize that both equations share the exact same roots, the path forward becomes clear. If two quadratic equations share both roots, they are essentially the same equation, scaled by a constant factor.

This leads us to the condition of proportionality:

$$\frac{a}{1} = \frac{b}{2} = \frac{c}{3}$$

This simple, symmetric relationship is the key to the entire problem. By setting these ratios equal, we immediately see that the ratio $a : b : c$ must be **$1 : 2 : 3$**.

It is a stunning example of how a seemingly complex problem about roots collapses into a simple, elegant ratio when you understand the underlying structure. Whenever you see common roots, always check the discriminant first; it is the compass that guides you through the storm of algebra.

Animated Solution for Mathematics - Quadratic Equations: If the equations $x^{2} + 2 x + 3 = 0$ and $a x^{2} + b x + c = 0, a, b, c \in R$ , have a common root, then $a : b : c$ is

Select Answer:

Visualized Solution

Analyze $x^{2} + 2 x + 3 = 0$

Discriminant Formula

Substitute Coefficients

Calculate $D$

Nature of Roots

Conjugate Pair Theorem

Second Equation

Both Roots Common

Condition for Common Roots

Apply Proportion

Final Ratio

The Sigma Insight: Common Roots

Analyzing the Parabola

The Power of the Conjugate Pair

The Elegance of Proportionality

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the equations x2+2x+3=0 and ax2+bx+c=0,a,b,c∈R, have a common root, then a:b:c is