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

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

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 $c$ .

${(- 4)}^{2} - 4 (1 \cdot 3)$

Evaluate the result to find the discriminant.

Simplify each term.

Raise $- 4$ to the power of $2$ .

$16 - 4 (1 \cdot 3)$

Multiply $3$ by $1$ .

$16 - 4 \cdot 3$

Multiply $- 4$ by $3$ .

$16 - 12$

Subtract $12$ from $16$ .

$4$

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