# How do you describe the nature of the roots of the equation 2x^2+3x=-3?

Jul 15, 2017

Roots are two complex numbers, conjugate of each other.

#### Explanation:

$2 {x}^{2} + 3 x = - 3$ can be written as $2 {x}^{2} + 3 x + 3 = 0$. We may now compare it with general equation $a {x}^{2} + b x + c = 0$

then Its discriminant is ${b}^{2} - 4 a c$ and for $2 {x}^{2} + 3 x + 3 = 0$ discriminant is ${3}^{2} - 4 \times 2 \times 3 = 9 - 24 = - 15$

As the discriminant is negative, but coefficients of equation are real

te roots of the equation are two complex numbers, conjugate of each other.

The roots are $- \frac{b}{2 a} \pm \frac{\sqrt{{b}^{2} - 4 a c}}{2 a}$

= $- \frac{3}{4} \pm \frac{\sqrt{- 15}}{4}$

i.e. $- \frac{3}{4} + i \frac{\sqrt{15}}{4}$ and $- \frac{3}{4} - i \frac{\sqrt{15}}{4}$

Jul 15, 2017

$\text{roots are not real}$

#### Explanation:

$\text{to determine the nature of the roots of a quadratic}$

$\text{use the "color(blue)"discriminant}$

•color(white)(x)Delta=b^2-4ac

• " if "Delta>0" the roots are real"

• " if "Delta=0" the roots are real and equal"

• " if "Delta<0" the roots are not real"

$\text{rearrange "2x^2+3x=-3" into standard form}$

$\text{that is } a {x}^{2} + b x + c = 0$

$\text{add 3 to both sides}$

$\Rightarrow 2 {x}^{2} + 3 x + 3 = 0$

$\text{with " a=2,b=3" and } c = 3$

$\Rightarrow \Delta = {3}^{2} - \left(4 \times 2 \times 3\right) = 9 - 24 = - 15$

$\text{since "Delta<0" then roots are not real}$