Quadratic Equation from Sum and Product of Roots

This problem asks us to find the specific quadratic equation when we know the sum and the product of its roots.

Understanding Quadratic Equations and Roots

A standard quadratic equation has the form: \(ax^2 + bx + c = 0\), where \(a \neq 0\).

For such an equation, the relationships between the coefficients (\(a, b, c\)) and the roots (let's call them \(\alpha\) and \(\beta\)) are:

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

A simpler way to write a quadratic equation, especially when the leading coefficient (\(a\)) is 1, is by using the sum and product directly:

Applying the Given Values

In this question, we are given:

• Sum of roots = 2
• Product of roots = -100

Now, we substitute these values into the formula:

Simplifying this equation, we get:

Analyzing the Options

Let's check the given options based on our derived equation:

• Option 1: "There are infinitely many such equations having different roots." This is incorrect. While we can multiply the entire equation by any non-zero constant (e.g., \(2x^2 - 4x - 200 = 0\)), the fundamental relationship defined by the sum and product leads to a unique standard form of the equation (\(x^2 + (b/a)x + c/a = 0\)). The roots themselves are also uniquely determined (up to order) by the sum and product.
• Option 2: "There is only one such equation which is \(x^2 + 2x - 100 = 0\)." If this were the equation, the sum of the roots would be \(-(2)/1 = -2\), not 2. So, this is incorrect.
• Option 3: "There is only one such equation which is \(x^2 - 2x - 100 = 0\)." This matches the equation we derived using the formula. The sum of roots is \(-(-2)/1 = 2\), and the product of roots is \(-100/1 = -100\). This matches the given information.
• Option 4: "There is no such equation." This is incorrect because we have successfully constructed the equation.

Conclusion

Based on the standard form of a quadratic equation derived from the sum and product of its roots, the correct equation is \(x^2 - 2x - 100 = 0\). This confirms that there is indeed only one such unique standard quadratic equation.