# How do you find the value of the discriminant and determine the nature of the roots x^2 + 4x = -7?

Sep 26, 2016

Roots are complex conjugates. For given equation they are $x = - 2 - 3 i$ or $x = - 2 + 3 i$

#### Explanation:

The discriminant of a quadratic equation $a {x}^{2} + b x + c = 0$ is ${b}^{2} - 4 a c$, which decides the nature of roots of the equation.

If $a$, $b$ and $c$ are rational and ${b}^{2} - 4 a c$ is square of a rational number, roots are rational.

If ${b}^{2} - 4 a c > 0$ but is not a square of a rational number, roots are real but not rational.

If ${b}^{2} - 4 a c > 0 - 0$ we have equal roots.

If ${b}^{2} - 4 a c < 0$ roots are complex, and if $a$, $b$ and $c$ are rational. they are complex conjugates

In ${x}^{2} + 4 x = - 7 \Leftrightarrow {x}^{2} + 4 x + 7 = 0$

the discriminant is ${4}^{2} - 4 \times 1 \times 7 = 16 - 28 = - 12$

hence roots are complex conjugates.

In fact ${x}^{2} + 4 x + 7 = 0$

$\Leftrightarrow {x}^{2} + 4 x + 4 - \left(- 3\right) = 0$

or ${\left(x + 2\right)}^{2} - \left(3 {i}^{2}\right) = 0$

or $\left(x + 2 + 3 i\right) \left(x + 2 - 3 i\right) = 0$

i.e. $x = - 2 - 3 i$ or $x = - 2 + 3 i$