Determine the Nature of the Roots Using the Discriminant 4x^2-8x+3=0

$4 x^{2} - 8 x + 3 = 0$

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 $c$ .

${(- 8)}^{2} - 4 (4 \cdot 3)$

Evaluate the result to find the discriminant.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$64 - 4 (4 \cdot 3)$

Multiply $4$ by $3$ .

$64 - 4 \cdot 12$

Multiply $- 4$ by $12$ .

$64 - 48$

Subtract $48$ from $64$ .

$16$

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