Determine the Nature of the Roots Using the Discriminant -4x^2+7x-2=0

$- 4 x^{2} + 7 x - 2 = 0$

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 $c$ .

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

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raise $7$ to the power of $2$ .

$49 - 4 (- 4 \cdot - 2)$

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

Multiply $- 4$ by $- 2$ .

$49 - 4 \cdot 8$

Multiply $- 4$ by $8$ .

$49 - 32$

Subtract $32$ from $49$ .

$17$

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 greater than $0$ , there are two real roots.

Two Real Roots