Determine the Nature of the Roots Using the Discriminant 6x^2+10x+4=0

$6 x^{2} + 10 x + 4 = 0$

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 $c$ .

$10^{2} - 4 (6 \cdot 4)$

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raise $10$ to the power of $2$ .

$100 - 4 (6 \cdot 4)$

Multiply $- 4 (6 \cdot 4)$ .

Multiply $6$ by $4$ .

$100 - 4 \cdot 24$

Multiply $- 4$ by $24$ .

$100 - 96$

Subtract $96$ from $100$ .

$4$

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