# What is the discriminant of x^2 -11x + 28 = 0 and what does that mean?

Jul 16, 2015

The discriminant is 9. It tells you that there are two real roots to the equation.

#### 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.

${x}^{2} - 11 x + 28 = 0$

Δ = b^2 – 4ac = 11^2 -4×1×28 = 121 – 112 = 9

This tells you that there are two real roots.

We can see this if we solve the equation.

${x}^{2} - 11 x + 28 = 0$

$\left(x - 7\right) \left(x - 4\right) = 0$

$\left(x - 7\right) = 0 \mathmr{and}$(x-4) = 0#

$x = 7$ or $x = 4$

There are two real roots to the equation.