Determine the Nature of the Roots Using the Discriminant 4x^2-4x+1=0

$4 x^{2} - 4 x + 1 = 0$

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 $c$ .

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

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

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

$16 - 4 (4 \cdot 1)$

Multiply $- 4 (4 \cdot 1)$ .

Multiply $4$ by $1$ .

$16 - 4 \cdot 4$

Multiply $- 4$ by $4$ .

$16 - 16$

Subtract $16$ from $16$ .

$0$

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

One Real Root