Find the sum and product of the roots of the following quadratic equations, and state their nature: a) x2 − 5x + 6 = 0 b) 3x2 − 12x + 12 = 0

Sum of roots $= -\frac{b}{a} = -\frac{-5}{1} = 5$

Product of roots $= \frac{c}{a} = \frac{6}{1} = 6$

$D = (-5)^2 - 4(1)(6) = 25 - 24 = 1$  
Since $D > 0$, the roots are **real and distinct**.

*   Sum of roots: $\boxed{5}$
*   Product of roots: $\boxed{6}$
*   Nature of roots: $\boxed{\text{Real and distinct}}$

Sum of roots $= -\frac{b}{a} = -\frac{-12}{3} = 4$

Product of roots $= \frac{c}{a} = \frac{12}{3} = 4$

$D = (-12)^2 - 4(3)(12) = 144 - 144 = 0$  
Since $D = 0$, the roots are **real and equal**.

*   Sum of roots: $\boxed{4}$
*   Product of roots: $\boxed{4}$
*   Nature of roots: $\boxed{\text{Real and equal}}$

$$
-1 + \beta = 6
$$

$$
\beta = 6 + 1
$$
$$
\beta = 7
$$

The other root is $\boxed{7}$.

$$
\text{Product} = \alpha \beta = (-1)(7)
$$
$$
\text{Product} = -7
$$

The product of the roots is $\boxed{-7}$.

$$
x^2 - (6)x + (-7) = 0
$$

$$
x^2 - 6x - 7 = 0
$$

The quadratic equation is $\boxed{x^2 - 6x - 7 = 0}$.