Determine the Nature of the Roots Using the Discriminant 3x^2-2x+5=0

$3 x^{2} - 2 x + 5 = 0$

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

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

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

${(- 2)}^{2} - 4 (3 \cdot 5)$

Evaluate the result to find the discriminant.

Simplify each term.

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

$4 - 4 (3 \cdot 5)$

Multiply $3$ by $5$ .

$4 - 4 \cdot 15$

Multiply $- 4$ by $15$ .

$4 - 60$

Subtract $60$ from $4$ .

$- 56$

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