Let's go through each of the quadratic equations to find the sum and product of the roots, which are based on the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

Formula for Sum and Product of Roots:

For a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are:

• Sum of the roots (S): $S = -\frac{b}{a}$
• Product of the roots (P): $P = \frac{c}{a}$

Now, applying these to each of the given equations:

(a) $x^2 + 3x + 2 = 0$

• Here, $a = 1$, $b = 3$, and $c = 2$.
• Sum of the roots: $S = -\frac{b}{a} = -\frac{3}{1} = -3$
• Product of the roots: $P = \frac{c}{a} = \frac{2}{1} = 2$

(b) $x^2 + mx + n = 0$

• In this general form, $a = 1$, $b = m$, and $c = n$.
• Sum of the roots: $S = -\frac{b}{a} = -\frac{m}{1} = -m$
• Product of the roots: $P = \frac{c}{a} = \frac{n}{1} = n$

(c) $4x^2 + 7x + 12 = 0$

• Here, $a = 4$, $b = 7$, and $c = 12$.
• Sum of the roots: $S = -\frac{b}{a} = -\frac{7}{4}$
• Product of the roots: $P = \frac{c}{a} = \frac{12}{4} = 3$

Would you like to go deeper into any of these steps or explore another topic related to quadratics?

Related Questions:

1. What is the quadratic formula, and how can it be used to find the roots directly?
2. How do the sum and product of the roots relate to the discriminant of a quadratic equation?
3. Can you derive the equation $x^2 + 3x + 2 = 0$ from its roots?
4. How do changes in coefficients $m$ and $n$ affect the roots of equation (b)?
5. What are the conditions for a quadratic equation to have real, equal, or complex roots?

Tip:

When working with quadratic equations, always check the discriminant $\Delta = b^2 - 4ac$ to determine the nature of the roots (real, complex, or equal).