Determine the Nature of the Roots Using the Discriminant x^2+2x+5=0

$x^{2} + 2 x + 5 = 0$

Step 1

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 2

Substitute in the values of $a$ , $b$ , and $c$ .

$2^{2} - 4 (1 \cdot 5)$

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raise $2$ to the power of $2$ .

$4 - 4 (1 \cdot 5)$

Multiply $- 4 (1 \cdot 5)$ .

Multiply $5$ by $1$ .

$4 - 4 \cdot 5$

Multiply $- 4$ by $5$ .

$4 - 20$

Subtract $20$ from $4$ .

$- 16$

Step 4

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant $(Δ)$ :

$Δ > 0$ means there are $2$ distinct real roots.

$Δ = 0$ means there are $2$ equal real roots, or $1$ distinct real root.

$Δ < 0$ means there are no real roots, but $2$ complex roots.

Since the discriminant is less than $0$ there are no real roots. Instead, there are two complex roots.

Two Complex Roots