We are given a quadratic equation \(x^2 + bx + c = 0\), where \(b > 0\) and \(c < 0\). Let the roots of this equation be \(\alpha\) and \(\beta\), with the condition that \(\alpha < \beta\).

Understanding Roots and Coefficients with Vieta's Formulas

For a general quadratic equation \(ax^2 + bx + c = 0\), Vieta's formulas relate the coefficients to the sum and product of the roots. For our equation \(x^2 + bx + c = 0\) (where \(a=1\)), the formulas are:

• Sum of roots: \(\alpha + \beta = -\frac{b}{1} = -b\)
• Product of roots: \(\alpha \beta = \frac{c}{1} = c\)

Analyzing the Given Conditions on Coefficients

We are given that \(b > 0\) and \(c < 0\).

• Since \(b > 0\), \(-b\) must be negative. Therefore, the sum of the roots \(\alpha + \beta = -b\) is negative, i.e., \(\alpha + \beta < 0\).
• Since \(c < 0\), the product of the roots \(\alpha \beta = c\) is negative, i.e., \(\alpha \beta < 0\).

The product of the roots being negative (\(\alpha \beta < 0\)) implies that the two roots must have opposite signs. One root is positive, and the other is negative. Given that \(\alpha < \beta\), \(\alpha\) must be the negative root and \(\beta\) must be the positive root.

Evaluating Statement 1: \(\alpha + \beta + \alpha\beta > 0\)

Let's substitute the expressions for the sum and product of roots from Vieta's formulas into the statement:

\(\alpha + \beta + \alpha\beta = (-b) + (c) = -b + c\)

We are given that \(b > 0\), which means \(-b\) is a negative number. We are also given that \(c < 0\), which means \(c\) is a negative number.

The sum of two negative numbers is always negative.

So, \(-b + c < 0\).

The statement \(\alpha + \beta + \alpha\beta > 0\) is equivalent to \(-b + c > 0\). Based on our analysis, this is false.

Therefore, Statement 1 is incorrect.

Evaluating Statement 2: \(\alpha^2\beta + \beta^2\alpha > 0\)

Let's factor the expression \(\alpha^2\beta + \beta^2\alpha\). We can factor out the common term \(\alpha\beta\):

\(\alpha^2\beta + \beta^2\alpha = \alpha\beta(\alpha + \beta)\)

Now, substitute the expressions for the product and sum of roots from Vieta's formulas:

\(\alpha\beta(\alpha + \beta) = (c)(-b) = -bc\)

We are given that \(b > 0\) and \(c < 0\).

The product \(bc\) will be the product of a positive number (\(b\)) and a negative number (\(c\)). The product of a positive and a negative number is always negative. So, \(bc < 0\).

Now consider \(-bc\). Since \(bc\) is negative, \(-bc\) will be the negative of a negative number, which is positive.

So, \(-bc > 0\).

The statement \(\alpha^2\beta + \beta^2\alpha > 0\) is equivalent to \(-bc > 0\). Based on our analysis, this is true.

Therefore, Statement 2 is correct.

Conclusion on the Correctness of Statements

Based on our evaluation:

• Statement 1 (\(\alpha + \beta + \alpha\beta > 0\)) is incorrect.
• Statement 2 (\(\alpha^2\beta + \beta^2\alpha > 0\)) is correct.

Thus, only Statement 2 is correct.

Quadratic Roots and Conditions Revision Table

Additional Information on Quadratic Equations and Roots

Understanding the relationship between the coefficients and the roots of a quadratic equation is fundamental. Vieta's formulas provide a direct link, allowing us to find the sum and product of roots without explicitly calculating the roots themselves.

Key points about roots based on coefficients:

• If the product of roots (\(\alpha \beta = c/a\)) is negative, the roots have opposite signs.
• If the product of roots is positive, the roots have the same sign. If the sum of roots (\(\alpha + \beta = -b/a\)) is also positive, both roots are positive. If the sum is negative, both roots are negative.
• The discriminant, \(\Delta = b^2 - 4ac\), tells us about the nature of the roots.
  • If \(\Delta > 0\), there are two distinct real roots.
  • If \(\Delta = 0\), there is exactly one real root (a repeated root).
  • If \(\Delta < 0\), there are no real roots (two complex conjugate roots).

In this problem, since \(\alpha\) and \(\beta\) are given as real roots, the discriminant \(b^2 - 4c\) must be greater than or equal to 0. Since \(b > 0\) (\(b^2 > 0\)) and \(c < 0\) (\(-4c > 0\)), \(b^2 - 4c\) is definitely positive, confirming two distinct real roots.