⁷This can be seen by finding the roots of the characteristic equation which are\[ \lambda =\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}=\frac{-k\pm \sqrt{k^{2}-4\omega _{n}^{2}}}{2}\] But \(\omega _{n}^{2}=\left ( \frac{vn\pi }{L}\right ) ^{2}\). Hence we can write the above as\[ \lambda =-\frac{k}{2}\pm \sqrt{\left ( \frac{k}{2}\right ) ^{2}-\omega _{n}^{2}}\] Hence the roots are both real.