Symmetric matrix real eigenvalues

[phrygian]

Homework Statement

Prove a symmetric (2x2) matrix always has real eigenvalues. The problem shows the matrix as {(a,b),(b,d)}.

Homework Equations

The problem says to use the quadratic formula.

The Attempt at a Solution

From the determinant I get (a-l)(d-l) - b^2 = 0 which expands to l^2 - (a+d)l + (ad - b^2) = 0

Using the quadratic formula I get for under the square root: (a + d)^2 - 4(ad-b^2)
How can I show that this is always positive?

Thanks for the help

 

[clamtrox]

You can write the square root term as a sum of squares, which is always positive.

 

[phrygian]

How do you deal with the -4ad term? I tried to factor it but couldn't figure out how

 

[clamtrox]

Can you see what (a+d)^2 - 4ad is?