(i)Express in the form \(4x^{2}-12x\) in the form \(\left ( 2x+a \right )^{2}+b\)

(ii)Hence ,or otherwise,find the set of values of x satisfying \(4x^{2}-12x> 7\)


(i)(\left ( 2x-2 \right )^{2}-9\)

(ii)\( 2x-3> 4\)    \(2x-3< – 4\)

\(x> 3\tfrac{1}{2}\) (or)  \(x< -\frac{1}{2}\)

Allow \(-\frac{1}{2}> x>3\tfrac{1}{2}\)


\(4x^{2}-12x-7\rightarrow \left ( 2x-7 \right )\left ( 2x+1 \right )\) 

\(x> 3\tfrac{1}{2}\)  (or) \(< -\frac{1}{2}\)

Allow \(-\frac{1}{2}> x>3\tfrac{1}{2}\)


1(i) Express \(x^{2}+6x+2\) in the form \(\left ( x+a \right )^{2}+b\),where a and b are constants.

(ii)Henc3,or otherwise, find the set of values of x for which \(x^{2}+6x+2> 9\).


(i)\(\left ( x+3 \right )^{2}-7\)

(ii)1,-7 seen

\(x> 1,x< -7\)


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

The function \(f:x\rightarrow -x^{2}+6x-5\) defined for \(x\geq m\) ,where m is constant.

(ii) State the smallest value of m for which f is one-one. 
(iii) For the case where m = 5, find an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\).



(ii)Smallest (m)-is 3


Correct order of operations 


Domain is\( x\leq 0\)



(i) Express \(x^{2}-2x-15\)  in the form \((x+a)^{2}+b\).

The function f is defined for p ≤ x ≤ q, where p and q are positive constants, by f : x → \(x^{2}\) − 2x − 15.
The range of f is given by c ≤ fx ≤ d, where c and d are constants.
(ii) State the smallest possible value of c. 
For the case where c = 9 and d = 65,
(iii) find p and q,
(iv) find an expression for \(f^{-1}(x)\).




(iii)\(9\leq (x-1)^{2}-16\leq 65\) OR \(x^{2}-2x-15=9\rightarrow 6,-4\)

\(25\leq (x-1)^{2}\leq 81 \) \(x^{2}-2x-15=65\rightarrow 10,-8\)

\(5\leq x-1\leq 9 \)             p=6

\(6\leq x\leq 10 \)        q=10

(iv)\(x=\left ( y-1 \right )^{2}-16\)      (interchange x/y)

\(y-1=\pm \sqrt{x+16}\)

f^{-1}\left ( x \right )=1+\sqrt{x+16}


 A curve for which \( \frac{\mathrm{d} y}{\mathrm{d} x}=7-x^{2}-6x \)   pases tghrough the p[oint (3,-10)

(i)Find the equation of the curve.

(ii)Express \(7-x^{2}-6x\) in the form \(a-(x+b)^{2}\),where a and b are constants.

(iii)Find the set of values of x for which the gradient of the curve is positive.



Uses (3,-10) →c=5


(iii)\(16-\left ( x+3 \right )^{2}> 0\rightarrow \left ( x+3 \right )^{2}< 16\),and solve

End-points x=1 or -7

\(\rightarrow -7< x< 1\)


The graph of y = f(x) is transformed to the graph of \(y = 1 + f(\frac{1}{2}x)\).
Describe fully the two single transformations which have been combined to give the resulting transformation.



[Stretch] [factor 2, x direction (or y-axis invariant)]
[Translation or Shift][1 unit in y direction] or
[Translation/Shift] \([\binom{0}{1}]\)


The function f is defined by \(f(x)=-2x^{2}+12x-3 for \varepsilon R\)

(i) Express \(-2x^{2}+12x-3 in the from 2(x+a)^{a}+b,\)where a and b are constants.

(ii) State the greatest value of fx.

The function g is defined by g(x) = 2x + 5 for x ∈ >.

(iii) Find the values of x for which gf(x )+ 1


= 0.

(i) \(-2(x-3) ^{2}\)+15(a=-3,b=15)

(ii) (f(x) ⩽) 15

(iii) gf(x) =\( (-2x^{2}+12x-3)+5=-4x^{2}+24x-6+5
gf(x)+1=0\rightarrow -4x^{2}+24x\)=0 x=0 or 6