Given the quadratic equation:

$x^2 + px + q = 0$

where $p$ and $q$ are constants, and the roots are $-3$ and $5$, we can use the relationships between the roots and coefficients of a quadratic equation.

For a quadratic equation $ax^2 + bx + c = 0$, with roots $\alpha$ and $\beta$, the relationships are:

1. Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
2. Product of the roots: $\alpha \cdot \beta = \frac{c}{a}$

In your case:

• $\alpha = -3$ and $\beta = 5$
• The equation is in the form $x^2 + px + q = 0$, so $a = 1$, $b = p$, and $c = q$.

Step 1: Find $p$

Using the sum of the roots:

$\alpha + \beta = -p$

Substitute $\alpha = -3$ and $\beta = 5$:

$-3 + 5 = -p$

This simplifies to:

$2 = -p$

Thus, $p = -2$.

Step 2: Find $q$

Using the product of the roots:

$\alpha \cdot \beta = q$

Substitute $\alpha = -3$ and $\beta = 5$:

$-3 \cdot 5 = q$

This simplifies to:

$q = -15$

Final answer:

• $p = -2$
• $q = -15$

Would you like more details on these steps?

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Here are 5 related questions to expand your understanding:

1. How can you verify the roots of a quadratic equation once you have found $p$ and $q$?
2. What is the general formula for solving a quadratic equation?
3. How does the discriminant of a quadratic equation help in determining the nature of its roots?
4. What happens to the sum and product of roots if the coefficients of the quadratic equation change?
5. How would the equation change if the roots were complex?

Tip: Always check your final result by plugging the values of $p$ and $q$ back into the original equation to ensure it holds true for the given roots.