Determine the Nature of the Roots Using the Discriminant 2x^2+5x-1=0

$2 x^{2} + 5 x - 1 = 0$

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 $c$ .

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

Evaluate the result to find the discriminant.

Simplify each term.

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

$25 - 4 (2 \cdot - 1)$

Multiply $2$ by $- 1$ .

$25 - 4 \cdot - 2$

Multiply $- 4$ by $- 2$ .

$25 + 8$

Add $25$ and $8$ .

$33$

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 greater than $0$ , there are two real roots.

Two Real Roots