Finding Parameter p for Negative Quadratic Roots

We are given the quadratic equation $x^2 - (p + 2)x + (2p+9)= 0$. For both roots to be negative real numbers, three conditions must be met:

• The discriminant ($\Delta$) must be non-negative ($\Delta \ge 0$).
• The sum of the roots must be negative.
• The product of the roots must be positive.

Applying Conditions to the Equation

For the equation $ax^2 + bx + c = 0$, we have $a=1$, $b=-(p+2)$, and $c=(2p+9)$.

1. Discriminant Condition ($\Delta \ge 0$)

Calculate the discriminant $\Delta = b^2 - 4ac$:

$\Delta = (-(p+2))^2 - 4(1)(2p+9)$

$\Delta = (p+2)^2 - 4(2p+9)$

$\Delta = (p^2 + 4p + 4) - (8p + 36)$

$\Delta = p^2 - 4p - 32$

We need $\Delta \ge 0$, so $p^2 - 4p - 32 \ge 0$. Factoring the quadratic gives:

$(p-8)(p+4) \ge 0$

This inequality holds when $p \le -4$ or $p \ge 8$. So, $p \in (-\infty, -4] \cup [8, \infty)$.

2. Sum of Roots Condition (< $-\mathbf{0}$>)

The sum of the roots is given by $-b/a$. In this case, the sum is $\frac{-(-(p+2))}{1} = p+2$. For negative roots, the sum must be negative:

$p+2 < 0 \implies p < -2$. So, $p \in (-\infty, -2)$.

3. Product of Roots Condition (> $\mathbf{0}$>)

The product of the roots is given by $c/a$. In this case, the product is $\frac{2p+9}{1} = 2p+9$. For negative roots, the product must be positive:

$2p+9 > 0 \implies 2p > -9 \implies p > -9/2$ or $p > -4.5$. So, $p \in (-4.5, \infty)$.

Combining Conditions and Finding the Interval

We need to find the values of $p$ that satisfy all three conditions simultaneously:

• $p \in (-\infty, -4] \cup [8, \infty)$
• $p \in (-\infty, -2)$
• $p \in (-4.5, \infty)$

Let's find the intersection:

1. Intersecting $(-\infty, -4] \cup [8, \infty)$ with $(-\infty, -2)$ gives $(-\infty, -4]$.
2. Intersecting $(-\infty, -4]$ with $(-4.5, \infty)$ gives $(-4.5, -4]$.

The set of all values for $p$ is the interval $(-4.5, -4]$.

Calculating $\beta - 2\alpha$

The problem states the interval is $(\alpha, \beta]$. Comparing this with our result $(-4.5, -4]$, we have:

• $\alpha = -4.5$
• $\beta = -4$

Now, we calculate $\beta - 2\alpha$:

$\beta - 2\alpha = -4 - 2(-4.5)$

= $-4 - (-9)$

= $-4 + 9$

= $5$