Determine the Nature of the Roots Using the Discriminant 3x^2-10=0

$3 x^{2} - 10 = 0$

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 $c$ .

$0^{2} - 4 (3 \cdot - 10)$

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$0 - 4 (3 \cdot - 10)$

Multiply $- 4 (3 \cdot - 10)$ .

Multiply $3$ by $- 10$ .

$0 - 4 \cdot - 30$

Multiply $- 4$ by $- 30$ .

$0 + 120$

Add $0$ and $120$ .

$120$

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