(a)

Express $x^{2}-8 x+11$ in the form $(x+p)^{2}+q$ where p and q are constants.

[ 2 ]

(b)

Hence find the exact solutions of the equation $x^{2}-8 x+11=1$ .

[ 2 ]

[Maximum number: 3]

Express $2 x^{2}-12 x+7$ in the form $a(x+b)^{2}+c$ , where a, b and c are constants.

[Maximum number: 5]

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

(a)

Express $x^{2}-4 x+8$ in the form $(x-a)^{2}+b$ .

[ 2 ]

(b)

Hence find the set of values of x for which f(x)<9, giving your answer in exact form.

[ 3 ]

(a)

Express $x^{2}+6 x+2$ in the form $(x+a)^{2}+b$ , where a and b are constants.

[ 2 ]

(b)

Hence, or otherwise, find the set of values of x for which $x^{2}+6 x+2>9$ .

[ 2 ]

[Maximum number: 3]

Express $3 x^{2}-12 x+7$ in the form $a(x+b)^{2}+c$ , where a, b and c are constants.

[Maximum number: 3]

Showing all necessary working, solve the equation $4 x-11 x^{\frac{1}{2}}+6=0$ .

(a)

Express $16 x^{2}-24 x+10$ in the form $(4 x+a)^{2}+b$ .

[ 2 ]

(b)

It is given that the equation $16 x^{2}-24 x+10=k$ , where k is a constant, has exactly one root.

Find the value of this root.

[ 2 ]

(a)

Express $x^{2}+6 x+5$ in the form $(x+a)^{2}+b$ , where a and b are constants.

[ 2 ]

(a)

Show that the equation

\log _{2}(x+5)=5-\log _{2} x

can be written as a quadratic equation in x.

[ 3 ]

(b)

Hence solve the equation

\log _{2}(x+5)=5-\log _{2} x

(a)

Given that $5^{2 x}+5^{x}=12$ , find the value of $5^{x}$ .

[ 3 ]