# How do you use the discriminant to determine the nature of the solutions given  9m^2 + 24m + 16 = 0?

Feb 5, 2017

$\text{roots are real and equal}$

#### Explanation:

$\text{For any quadratic equation "ax^2+bx+c=0" the nature of the roots depends on the discriminant } \Delta = {b}^{2} - 4 a c$

$\Delta > 0 \text{ roots are real and unequal}$

$\Delta = 0 \text{ roots are real and equal}$

$\Delta < 0 \text{ roots are complex}$

In this case:

$9 {m}^{2} + 24 m + 16 = 0$

$a = 9 , b = 24 , c = 16$

$\Delta = {24}^{2} - 4 \times 9 \times 16$

$\Delta = 576 - 576 = 0$

$\therefore \text{ roots are real and equal}$

this can be confirmed by factorising

$9 {m}^{2} + 24 m + 16 = 0 \implies {\left(3 m + 4\right)}^{2} = 0$