Determine the Nature of the Roots Using the Discriminant 4x^2+x-5=0

$4 x^{2} + x - 5 = 0$

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 $c$ .

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

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

One to any power is one.

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

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

Multiply $4$ by $- 5$ .

$1 - 4 \cdot - 20$

Multiply $- 4$ by $- 20$ .

$1 + 80$

Add $1$ and $80$ .

$81$

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

Two Real Roots