Determine the Nature of the Roots Using the Discriminant 0=3x^2-7x+4

$0 = 3 x^{2} - 7 x + 4$

Step 1

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

Subtract $3 x^{2}$ from both sides of the equation.

$- 3 x^{2} = - 7 x + 4$

Add $7 x$ to both sides of the equation.

$- 3 x^{2} + 7 x = 4$

Subtract $4$ from both sides of the equation.

$- 3 x^{2} + 7 x - 4 = 0$

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 $c$ .

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

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

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

$49 - 4 (- 3 \cdot - 4)$

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

Multiply $- 3$ by $- 4$ .

$49 - 4 \cdot 12$

Multiply $- 4$ by $12$ .

$49 - 48$

Subtract $48$ from $49$ .

$1$

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