steps.tex

$$
2 \lambda_{\pm}  = 
\alpha-t+2-\beta
\pm
\sqrt{
\beta^{2}
+
(\alpha - t)^{2}
+
2\beta ( t - \alpha )(|u|^2
-
|v|^2
)
}
$$

$$
2 \lambda_{\pm}  = 
\alpha-t+2-\beta
\pm
\sqrt{
\beta^{2}
+
(\alpha - t)^{2}
+
2 t \beta (|u|^2
-
|v|^2
)
-
2\beta \alpha( |u|^2
-
|v|^2}
)
$$

$$
2 \lambda_{\pm}  = 
\alpha-t+2-\beta
\pm
\sqrt{
\beta^{2}
+
(\alpha - t)^{2}
+
2 t \beta |u|^2
-
2 t \beta |v|^2
-
2\beta \alpha |u|^2
+
2 \alpha \beta |v|^2}
$$
$$
2 \lambda_{\pm}  = 
\alpha-t+2-\beta
\pm
\sqrt{
\beta^{2}
+
\underbrace{
\alpha^{2}
+
t^{2}
-
2 \alpha t
}
+
2 t \beta |u|^2
-
2 t \beta |v|^2
-
2\beta \alpha |u|^2
+
2 \alpha \beta |v|^2}
$$
$$
2 \lambda_{\pm}  = 
\alpha-t+2-\beta
\pm
\sqrt{
\alpha^{2}
+
\underbrace{
\beta^{2}|u|^4
+
\beta^{2}  |v|^4
+
2\beta^{2}|u|^{2}|v|^{2}
}
+
t^{2}
-
2 \alpha t
+
2 t \beta |u|^2
-
2 t \beta |v|^2
-
2\beta \alpha |u|^2
+
2 \alpha \beta |v|^2}
$$
$$
2 \lambda_{\pm}  = 
\alpha-t+2-\beta
\pm
\sqrt{
\alpha^{2}
+
4\beta^{2}|u|^{2}|v|^{2}
+
t^{2}
-
2 \alpha t
+
2 t \beta |u|^2
-
2 t \beta |v|^2
+
\beta^{2}|u|^4
-
2\beta \alpha |u|^2
-
2 \beta^{2} |u|^2 |v|^2
+
\beta^{2}  |v|^4
+
2 \alpha \beta |v|^2}
$$