# What is the discriminant of 5x^2 + 11x + 8 = 0 and what does that mean?

Jul 16, 2015

The discriminant is -39. 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.

$5 {x}^{2} + 11 x + 8 = 0$

Δ = b^2 – 4ac = 11^2 -4×5×8 = 121 – 160 = -39

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

We can see this if we solve the equation.

$5 {x}^{2} + 11 x + 8 = 0$

x = (-b±sqrt(b^2-4ac))/(2a) = (-11±sqrt(11^2 -4×5×8))/(2 ×5) = (-11±sqrt(121-160))/10 = (-11±sqrt(-39))/10 = 1/10(-11 ±isqrt39) = -1/10(11±isqrt39)

$x = - \frac{1}{10} \left(11 + i \sqrt{39}\right)$ and $x = - \frac{1}{10} \left(11 - i \sqrt{39}\right)$

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