# How do you find the value of the discriminant and determine the nature of the roots  2x^2+3x+1=0?

Oct 3, 2016

The Roots are Real, distinct and Rational...

So, go ahead and solve the equation, you will 2 exact answers for the roots!

#### Explanation:

The discriminant ($\Delta$) tells us something about the roots (solutions) of a quadratic equation without us having to solve the equation first.

$\Delta = {b}^{2} - 4 a c \text{ where } a {x}^{2} + b x + c = 0$

From $\text{ } 2 {x}^{2} + 3 x + 1 = 0$

$\Delta = {3}^{2} - 4 \left(2\right) \left(1\right) = 9 - 8 = 1$

$\Delta = 1$

What does this tell us?

If $\Delta < 0 \text{ } \rightarrow$ the roots are non-real (imaginary)

If $\Delta \ge 0 \text{ } \rightarrow$ the roots are Real. (they do exist!)

If $\Delta = 0 \text{ } \rightarrow$ there are 2 equal roots (ie one answer)

If $\Delta > 0 \text{ } \rightarrow$ there are 2 distinct Real roots. (different)

If $\Delta \text{is a square } \rightarrow$ the roots are Rational.

If $\Delta \text{is not a square } \rightarrow$ the roots are Irrational.

$1$ is a perfect square, so the Roots are Real, distinct and Rational.