# What is the discriminant of 3x^2 - 5x + 4 = 0 and what does that mean?

Jul 16, 2015

The discriminant is -23. It tells you that there are no real roots to the equation, but there are two complex roots.

#### Explanation:

If you have a quadratic equation of the form

$a {x}^{2} + b x + c = 0$

The solution is

x = (-b±sqrt(b^2-4ac))/(2a)

The discriminant Δ is ${b}^{2} - 4 a c$.

The discriminant "discriminates" the nature of the roots.

There are three possibilities.

• If Δ > 0, there are two separate real roots.
• If Δ = 0, there are two identical real roots.
• If Δ <0, there are no real roots, but there are two complex roots.

3x^2 – 5x +4 = 0

Δ = b^2 – 4ac = (-5)^2 -4×3×4 = 25 – 48 = -23

This tells you that there are no real roots, but there are two complex roots.

We can see this if we solve the equation.

3x^2 – 5x +4 = 0

x = (-b±sqrt(b^2-4ac))/(2a) = (-(-5)±sqrt((-5)^2 -4×3×4))/(2 ×3) = (5±sqrt(25-48))/6 = (5±sqrt(-23))/6 = 1/6(5 ±isqrt23)

$x = \frac{1}{6} \left(5 + i \sqrt{23}\right)$ and $x = \frac{1}{6} \left(5 - i \sqrt{23}\right)$

There are no real roots, but there are two complex roots to the equation.