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

$4 x^{2} = 2 x - 4$

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$4 x^{2} - 2 x = - 4$

Add $4$ to both sides of the equation.

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

Step 2

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 3

Substitute in the values of $a$ , $b$ , and $c$ .

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

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

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

$4 - 4 (4 \cdot 4)$

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

Multiply $4$ by $4$ .

$4 - 4 \cdot 16$

Multiply $- 4$ by $16$ .

$4 - 64$

Subtract $64$ from $4$ .

$- 60$

Step 5

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 less than $0$ there are no real roots. Instead, there are two complex roots.

Two Complex Roots