Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

Differentiate $ \sin(ax+b) e ^ {nx}$.

Differentiate $x^m\cos(ax+b)$

Differentiate $ (a+bx) ^ {m} \sin(nx)$

Differentiate $ (ax+b)^m(cx+d)^n$ using the product rule. The answer will be of the form $(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.

Differentiate $ x ^ m(ax+b)^n$ using the product rule. The answer will be of the form $x^{m-1}(ax+b)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.

The derivative of  $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

Differentiate $f(x) = x^m(a x+b)^n$.

Differentiate $ x ^m \sqrt{a x+b}$.
The answer is in the form $\displaystyle \frac{x^{m-1}g(x)}{2\sqrt{ax+b}}$
for a polynomial $g(x)$. Find $g(x)$.

Differentiate the function $(a + b x)^m  e ^ {n x}$ using the product rule.

Differentiate the function $f(x)=(a + b x)^m  e ^ {n x}$ using the product rule. Find $g(x)$ such that $f\;'(x)= (a + b x)^{m-1}  e ^ {n x}g(x)$.

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$. 

Hence, or otherwise, find roots of  $ax^2+bx+c=0$.

Includes a video which shows how to solve a quadratic by completing the square.

Differentiate $\displaystyle \ln((ax+b)^{m})$

Differentiate

\[ \sqrt{a x^m+b})\]

Differentiate $\displaystyle e^{ax^{m} +bx^2+c}$

Differentiate $\displaystyle \cos(e^{ax}+bx^2+c)$.

Contains a video solving a similar example.

Factorise $\displaystyle{ax ^ 2 + bx + c}$ into linear factors. 

Find $c$ and $d$ such that $x^2+ax+b = (x+c)^2+d$.

Find $\displaystyle I=\int \frac{2 a x + b} {a x ^ 2 + b x + c}\;dx$ by substitution or otherwise.

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.

A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.

There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$

Find $\displaystyle \int \frac{c}{\sqrt{a-bx^2}}\;dx$. Solution involves the inverse trigonometric function $\arcsin$.

Find the polynomial $g(x)$ such that $\displaystyle \int \frac{ax+b}{(cx+d)^{n}} dx=\frac{g(x)}{(cx+d)^{n-1}}+C$.

Integrate $f(x) = ae ^ {bx} + c\sin(dx) + px^q$. Must input $C$ as the constant of integration.

Find (hyperbolic substitution):
$\displaystyle \int_{b}^{2b} \left(\frac{1}{\sqrt{a^2x^2-b^2}}\right)\;dx$

Given $\displaystyle \int (ax+b)e^{cx}\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\displaystyle \int (ax+b)^2e^{cx}\;dx =h(x)e^{cx}+C$. 

Find $\displaystyle \int (ax+b)\cos(cx+d)\; dx $ and hence find $\displaystyle \int (ax+b)^2\sin(cx+d)\; dx $