Determine the Nature of the Roots Using the Discriminant 16x^2-16x+4=0

$16 x^{2} - 16 x + 4 = 0$

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 $c$ .

${(- 16)}^{2} - 4 (16 \cdot 4)$

Evaluate the result to find the discriminant.

Simplify each term.

Raise $- 16$ to the power of $2$ .

$256 - 4 (16 \cdot 4)$

Multiply $16$ by $4$ .

$256 - 4 \cdot 64$

Multiply $- 4$ by $64$ .

$256 - 256$

Subtract $256$ from $256$ .

$0$

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 equal to $0$ , there are two equal roots, or one distinct real root.

One Real Root