Determine the Nature of the Roots Using the Discriminant -5x^2+9x-3=0

$- 5 x^{2} + 9 x - 3 = 0$

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 $c$ .

$9^{2} - 4 (- 5 \cdot - 3)$

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raise $9$ to the power of $2$ .

$81 - 4 (- 5 \cdot - 3)$

Multiply $- 4 (- 5 \cdot - 3)$ .

Multiply $- 5$ by $- 3$ .

$81 - 4 \cdot 15$

Multiply $- 4$ by $15$ .

$81 - 60$

Subtract $60$ from $81$ .

$21$

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