Determine the Nature of the Roots Using the Discriminant x^2+5x+11=0

$x^{2} + 5 x + 11 = 0$

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 $c$ .

$5^{2} - 4 (1 \cdot 11)$

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raise $5$ to the power of $2$ .

$25 - 4 (1 \cdot 11)$

Multiply $- 4 (1 \cdot 11)$ .

Multiply $11$ by $1$ .

$25 - 4 \cdot 11$

Multiply $- 4$ by $11$ .

$25 - 44$

Subtract $44$ from $25$ .

$- 19$

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