Determine the Nature of the Roots Using the Discriminant 8x+2=-8x^2

$8 x + 2 = - 8 x^{2}$

Step 1

Add $8 x^{2}$ to both sides of the equation.

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

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 $c$ .

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

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

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

$64 - 4 (8 \cdot 2)$

Multiply $- 4 (8 \cdot 2)$ .

Multiply $8$ by $2$ .

$64 - 4 \cdot 16$

Multiply $- 4$ by $16$ .

$64 - 64$

Subtract $64$ from $64$ .

$0$

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

One Real Root