If α and β (α

Q: Wait, why do we reject the possibility that both roots are negative?

That is a great question! Look at the product of the roots, $\alpha\beta = c$. We are given that $c < 0$. If both roots were negative, their product would be positive (a negative times a negative is a positive). Since the product is negative, the roots must have opposite signs.

Q: Why does $b > 0$ force the sum to be negative?

Remember Vieta's formula: $\alpha + \beta = -\frac{b}{a}$. Since our leading coefficient $a$ is $1$, the sum is simply $-b$. If $b$ is a positive number, then $-b$ must be negative. Therefore, the sum of the roots is strictly less than zero.

Q: How do I interpret $\beta < -\alpha$ as an absolute value?

Since we established that $\alpha$ is the negative root, its absolute value is defined as $|\alpha| = -\alpha$. So, when we derive $\beta < -\alpha$, we are essentially saying that the positive root $\beta$ is closer to the origin than the negative root $\alpha$ is. It is a beautiful way to compare their distances from zero!

Question

If $\alpha$ and $\beta$ ($\alpha < \beta$) are the roots of the equation $x^2 + bx + c = 0$, where $c < 0 < b$, then

Options:

(A) $0 < \alpha < \beta$

(B) $\alpha < 0 < \beta < |\alpha|$

(C) $\alpha < \beta < 0$

(D) $\alpha < 0 < |\alpha| < \beta$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a graph of the function $f(x) = x^2 + bx + c$. Because the coefficient of $x^2$ is positive, this parabola opens upwards, like a giant, welcoming smile.

The roots, $\alpha$ and $\beta$, are the points where this smile crosses the x-axis. This is our starting point—a geometric reality that anchors our algebraic journey.

### The Keys to the Kingdom: Vieta's Formulas

To unlock the secrets of these roots, we turn to the legendary Vieta's formulas. They are the bridge between the roots and the coefficients.

We know that the product of the roots is:

$$\alpha\beta = c$$

The sum of the roots is:

$$\alpha + \beta = -b$$

These two simple equations are the master keys that will reveal the nature of $\alpha$ and $\beta$.

### The Dance of Signs

Let's look at the product first. The problem tells us that $c < 0$. Since $\alpha\beta = c$, it follows that $\alpha\beta < 0$.

In the world of numbers, the only way for a product to be negative is if the two numbers have opposite signs. One must be positive, and one must be negative.

Given the condition $\alpha < \beta$, we can immediately place them on the number line: $\alpha$ must be the negative root, and $\beta$ must be the positive root. Thus, we have $\alpha < 0 < \beta$.

### The Tug-of-War: Analyzing the Sum

Now, let's consider the sum. We are given $b > 0$, which means $-b < 0$. Therefore:

$$\alpha + \beta < 0$$

Think about this: we are adding a negative number $\alpha$ and a positive number $\beta$, and the result is still negative. This implies that the negative root $\alpha$ has a larger magnitude—a stronger 'pull'—than the positive root $\beta$.

Mathematically, $\alpha + \beta < 0$ rearranges to:

$$\beta < -\alpha$$

### The Final Reveal

We are almost there! We know that $\alpha$ is negative, so its absolute value is $|\alpha| = -\alpha$. Substituting this into our inequality $\beta < -\alpha$, we get:

$$\beta < |\alpha|$$

This tells us that the distance from the origin to the positive root $\beta$ is smaller than the distance from the origin to the negative root $\alpha$.

Combining everything we have discovered, we arrive at the elegant conclusion:

**$$\alpha < 0 < \beta < |\alpha|$$**

You have just navigated the interplay between algebra and geometry, proving that even in a simple quadratic, there is a deep, structured beauty waiting to be uncovered. Keep exploring, keep questioning, and let the math guide your intuition!

Animated Solution for Mathematics - Quadratic Equations: If $α$ and $β$ ( $α < β$ ) are the roots of the equation $x^{2} + b x + c = 0$ , where $c < 0 < b$ , then

Select Answer:

Visualized Solution

Visualizing the Quadratic Equation

Applying Vieta's Formulas

Analyzing the Product of Roots

Determining the Signs of the Roots

Ordering the Roots

Analyzing the Sum of Roots

Implications of a Negative Sum

Introducing Absolute Value

Comparing Magnitudes

Final Conclusion

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Keys to the Kingdom

The Dance of Signs

The Tug-of-War

The Final Reveal

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