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

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

Step 1

Subtract $7$ from both sides of the equation.

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

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 $c$ .

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

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

Raise $3$ to the power of $2$ .

$9 - 4 (3 \cdot - 7)$

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

Multiply $3$ by $- 7$ .

$9 - 4 \cdot - 21$

Multiply $- 4$ by $- 21$ .

$9 + 84$

Add $9$ and $84$ .

$93$

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

Two Real Roots