In a quadratic equation of the form $ax^2 + bx + c = 0$, the sum and product of the roots can be related to the coefficients using Vieta's formulas:

• Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
• Product of the roots: $\alpha \beta = \frac{c}{a}$

Given that the two roots are equal and opposite in sign, we can denote the roots as $\alpha = r$ and $\beta = -r$.

Step 1: Sum of the roots

The sum of the roots is: $\alpha + \beta = r + (-r) = 0$

From Vieta's formula, we know: $\alpha + \beta = -\frac{b}{a}$ So, we set: $0 = -\frac{b}{a}$ This implies: $b = 0$

Conclusion:

The value of $b$ is $0$.

Let me know if you'd like further clarification or if you have any other questions!

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Related Questions:

1. How does Vieta's formula work in general for any quadratic equation?
2. What happens if the roots of a quadratic equation are real but not equal?
3. Can we derive the quadratic equation if the sum and product of the roots are given?
4. What is the discriminant, and how does it relate to the nature of the roots?
5. What is the significance of the coefficients in determining the nature of the roots of a quadratic equation?

Tip: When the roots of a quadratic equation are equal and opposite, it indicates symmetry around the y-axis, making the equation symmetrical in its graph as well.