Determine the Nature of the Roots Using the Discriminant 7x^2+10x+3=0

$7 x^{2} + 10 x + 3 = 0$

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 $c$ .

$10^{2} - 4 (7 \cdot 3)$

Step 3

Evaluate the result to find the discriminant.

Simplify each term.

Raise $10$ to the power of $2$ .

$100 - 4 (7 \cdot 3)$

Multiply $- 4 (7 \cdot 3)$ .

Multiply $7$ by $3$ .

$100 - 4 \cdot 21$

Multiply $- 4$ by $21$ .

$100 - 84$

Subtract $84$ from $100$ .

$16$

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