Determine the Nature of the Roots Using the Discriminant 3x^2-8x+5=5x^2

$3 x^{2} - 8 x + 5 = 5 x^{2}$

Move all terms to the left side of the equation and simplify.

Subtract $5 x^{2}$ from both sides of the equation.

$3 x^{2} - 8 x + 5 - 5 x^{2} = 0$

Subtract $5 x^{2}$ from $3 x^{2}$ .

$- 2 x^{2} - 8 x + 5 = 0$

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Substitute in the values of $a$ , $b$ , and $c$ .

${(- 8)}^{2} - 4 (- 2 \cdot 5)$

Evaluate the result to find the discriminant.

Simplify each term.

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

$64 - 4 (- 2 \cdot 5)$

Multiply $- 2$ by $5$ .

$64 - 4 \cdot - 10$

Multiply $- 4$ by $- 10$ .

$64 + 40$

Add $64$ and $40$ .

$104$

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