Quadratic inequalities - Completing the square and sketching

5 marks Easy -1.2

9

Show that $( x - 1 ) ( x - 2 ) ( x - 3 ) - \left( x ^ { 3 } - x ^ { 2 } + 11 x - 12 \right) = 6 - 5 x ^ { 2 }$.
Solve the equation $6 - 5 x ^ { 2 } = 0$.

11 marks Standard +0.3

7

1. Express $4 - 10 x - x ^ { 2 }$ in the form $p - ( x + q ) ^ { 2 }$.
2. Hence write down the equation of the line of symmetry of the curve with equation $y = 4 - 10 x - x ^ { 2 }$.
The curve $C$ has equation $y = 4 - 10 x - x ^ { 2 }$ and the line $L$ has equation $y = k ( 4 x - 13 )$, where $k$ is a constant.
1. Show that the $x$-coordinates of any points of intersection of the curve $C$ with the line $L$ satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
2. Given that the curve $C$ and the line $L$ intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
3. Solve the inequality $4 k ^ { 2 } + 33 k + 29 > 0$.

5 marks Moderate -0.8

The function f is defined by $\text{f}(x) = x^2 - 4x + 8$ for $x \in \mathbb{R}$.

Express $x^2 - 4x + 8$ in the form $(x - a)^2 + b$. [2]
Hence find the set of values of $x$ for which $\text{f}(x) < 9$, giving your answer in exact form. [3]

6 marks Moderate -0.3

Write $3x^2 - 6x + 1$ in the form $p(x + q)^2 + r$, where $p$, $q$ and $r$ are integers. [2]
Solve $3x^2 - 6x + 1 \leq 0$, giving your answer in set notation. [4]