mat 1996 Q2

mat · Uk Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system

2. (a) Factorise the expression $x ^ { 2 } + x - 6$.
(b) For which values of the real constant $a$ does the equation
$$x ^ { 2 } + x - a = 0$$
have at least one real solution? Write down these solutions in terms of $a$.
(c) Show that, for any value of the real constant $b$, the equation
$$x ^ { 3 } - ( b + 1 ) x + b = 0$$
has $x = 1$ as a solution. Find all values of $b$ for which this equation has exactly two distinct solutions.

2. (a) Factorise the expression $x ^ { 2 } + x - 6$.\\
(b) For which values of the real constant $a$ does the equation

$$x ^ { 2 } + x - a = 0$$

have at least one real solution? Write down these solutions in terms of $a$.\\
(c) Show that, for any value of the real constant $b$, the equation

$$x ^ { 3 } - ( b + 1 ) x + b = 0$$

has $x = 1$ as a solution. Find all values of $b$ for which this equation has exactly two distinct solutions.\\

Paper Questions

Q2 Q3 Q4 Q5