Determine the Nature of the Roots Using the Discriminant 1/2x^2+4x+8=0

$\frac{1}{2} x^{2} + 4 x + 8 = 0$

Step 1

Simplify the left side.

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} + 4 x + 8 = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (4 x) + 2 \cdot 8 = 0$

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (4 x) + 2 \cdot 8 = 0$

Rewrite the expression.

$x^{2} + 2 (4 x) + 2 \cdot 8 = 0$

Multiply $4$ by $2$ .

$x^{2} + 8 x + 2 \cdot 8 = 0$

Multiply $2$ by $8$ .

$x^{2} + 8 x + 16 = 0$

Step 3

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

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

Step 4

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

$8^{2} - 4 (1 \cdot 16)$

Step 5

Evaluate the result to find the discriminant.

Simplify each term.

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

$64 - 4 (1 \cdot 16)$

Multiply $- 4 (1 \cdot 16)$ .

Multiply $16$ by $1$ .

$64 - 4 \cdot 16$

Multiply $- 4$ by $16$ .

$64 - 64$

Subtract $64$ from $64$ .

$0$

Step 6

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