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Solving a pair of simultaneous equations of the form $y=mx+c_1$ and $y=ax^2+kx+c_2$ to find the possible values for the unknown coefficient $k$, when given the values of $m$, $a$, $c_1$ and $c_2$.

Solving a pair of simultaneous equations of the form $a_1x+y=c_1$ and $a_2x^2+b_2xy=c_2$ by forming a quadratic equation.

Solving a quadratic equation via factorisation (or otherwise) with the $x^2$-term having a coefficient of 1.

Solving a quadratic equation of the form $ax^2+bx+c=0$ using the quadratic formula.

Determining the number of real roots a quadratic equation has by evaluating and interpreting the discriminant.

Part (a): Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, give an expression for the function $f(x)$, where $f(x)=g(x)-2h(x)$. 

Part (b): Solve $f(x)=0$.

Given two cubic functions $g(x)$ and $h(x)$ of the form $ax^3+bx^2+cx+d$, solve the equation $g(x)=2h(x)$, giving all possible solutions for $x$. 

Given a cubic expression and a linear factor, calculate the remaining quadratic factor.

Calculating two unknown coefficients of a cubic expression when given two of the expression's linear factors.

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