Determine the Nature of the Roots Using the Discriminant -4x^2-2x-1=0

$- 4 x^{2} - 2 x - 1 = 0$

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 $c$ .

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

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

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

$4 - 4 (- 4 \cdot - 1)$

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

Multiply $- 4$ by $- 1$ .

$4 - 4 \cdot 4$

Multiply $- 4$ by $4$ .

$4 - 16$

Subtract $16$ from $4$ .

$- 12$

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