I am trying to derive the following formula given by the lecture notes
$$SE(\hat{\beta_0}+\hat{\beta_1}x_0)=\hat{\sigma}\bigg[\frac{1}{n}+\frac{(x_0-\bar{x})^2}{(n-1)s^2_x}\bigg]^\frac{1}{2}$$
My attempt
$$SE(\hat{\beta_0}+\hat{\beta_1}x_0)=\big[SE^2(\hat{\beta_0})+x_0^2SE^2(\hat{\beta_1})\big]^\frac{1}{2}\\ SE^2(\hat{\beta_0})=\hat{\sigma}^2\bigg[\frac{1}{n}+\frac{\bar{x}^2}{(n-1)s^2_x}\bigg]\\ x^2_0SE^2(\hat{\beta_1})=\hat{\sigma}^2\bigg[\frac{x^2_0 }{(n-1)s^2_x}\bigg]\\ \text{Plug in}\\ SE(\hat{\beta_0}+\hat{\beta_1}x_0)=\hat{\sigma}\bigg[\frac{1}{n}+\frac{\bar{x}^2}{(n-1)s^2_x}+\frac{x^2_0}{(n-1)s^2_x}\bigg]^{\frac{1}{2}}=\hat{\sigma}\bigg[\frac{1}{n}+\frac{\color{red}{x^2_0+\bar{x}^2}}{(n-1)s^2_x}\bigg]^{\frac{1}{2}} $$
In the correct answer, the numerator should be $(x_0-\bar{x})^2$, I don't see how can I obtained that.
Any comment are more than welcome and appreciated it for that.