By sketching a suitable pair of graphs, show that the equation $e^{2x+1}=14-x^3$ has exactly one real root. Show by calculation that this root lies between $0.5$ and $1$.

The equation $\frac{\ln x^2-6}{2}=x-5$ has a root $\alpha ,$ such that $p<\alpha <q,$ where $p$ and $q$ are consecutive integers.

Show that $\alpha$ also satisfies the equation $x=\ln x+2.$

The equation $\frac{\ln x^2-6}{2}=x-5$ has a root $\alpha ,$ such that $p<\alpha <q,$ where $p$ and $q$ are consecutive integers.

Find the value of $p$ and the value of $q.$

The equation $\frac{\ln x^2-6}{2}=x-5$ has a root $\alpha ,$ such that $3<\alpha <4$. Use the iterative formula based on the equation $x=\ln x+2$ with starting value $x=3$ to find the value of $\alpha$ correct to $3$ decimal places.

Use iterative formula based on the equation $5^y-2=y\ln 5$$\left(\alpha \text{is a root}\right)$ with a starting value of $y=0.5$ to find the value of $\alpha$ correct to two decimals. Use $\left(\ln 5=1.609437912\right)$