Determine the Nature of the Roots Using the Discriminant 6x^2=-8x-7

$6 x^{2} = - 8 x - 7$

Step 1

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

Add $8 x$ to both sides of the equation.

$6 x^{2} + 8 x = - 7$

Add $7$ to both sides of the equation.

$6 x^{2} + 8 x + 7 = 0$

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 $c$ .

$8^{2} - 4 (6 \cdot 7)$

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

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

$64 - 4 (6 \cdot 7)$

Multiply $- 4 (6 \cdot 7)$ .

Multiply $6$ by $7$ .

$64 - 4 \cdot 42$

Multiply $- 4$ by $42$ .

$64 - 168$

Subtract $168$ from $64$ .

$- 104$

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