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

##### 1 Answer
Jul 16, 2015

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

Your equation is

2x^2 – 3x +4 = 0

Δ = b^2 – 4ac = (-3)^2 -4×2×4 = 9 – 32 = -23

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

We can see this if we solve the equation.

2x^2–3x+4 = 0

x = (-b±sqrt(b^2-4ac))/(2a) = (-(-3)±sqrt((-3)^2 -4×2×4))/(2×2) = (3±sqrt(9-32))/4 = (3±sqrt(-23))/4 = 1/4(3±isqrt23)

$x = \frac{1}{4} \left(3 + i \sqrt{23}\right)$ and $x = \frac{1}{4} \left(3 - i \sqrt{23}\right)$

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