```Higher Quadratic
Past Paper Questions
2001 Paper I Q2
For what values of k does the equation x2 – 5x + (k + 6) = 0 have equal roots?
a=1
b=–5
c = (k + 6)
1x2 – 5x + (k + 6) = 0
Equal roots 
b2 – 4ac = 0
(– 5)2 – 4(1)(k + 6) = 0
25 – 4k – 24 = 0
1 – 4k = 0
1 = 4k
k=¼
3
2002 Paper 2 Q9
Show that the equation (1 – 2k)x2 – 5kx – 2k = 0 has real roots for all integer
values of k?
a = 1 – 2k
(1 – 2k)x2 – 5kx – 2k = 0
b = – 5k
c = - 2k
Real roots  b2 – 4ac ≥ 0
(– 5k)2 – 4(1 – 2k)(– 2k) ≥ 0
25k2 + 8k(1 – 2k) ≥ 0
25k2 + 8k – 16k2 ≥ 0
9k2 + 8k ≥ 0
For all integers 9k2 ≥ 0 & 9k2 ≥ 8k
 always real roots for all integer values of k.
5
2003 Paper 1 Q7
Show that the line with equation y = 2x + 1 does not intersect the parabola with
equation y = x2 + 3x + 4
a=1
b=1
c=3
x2 + 3x + 4 = 2x + 1
x2 + x + 3 = 0
Find value of b2 – 4ac
(1)2 – 4(1)(3) = 1 – 12 = -11 < 0
As b2 – 4ac < 0  No intersection occurs.
5
2005 Paper 2 Q11
(a) Show that x = -1 is a solution of the cubic x3 + px2 + px + 1 = 0
(b) Hence find the range of values of p for which all the roots are real
1
7
x3 + px2 + px + 1 = 0
1
-1
p
-1
1 P–1
p
1
1
0
1–p -1
As Remainder, R = 0  x = -1 is a solution
&
x3 + px2 + px + 1 = (x + 1)(1x2 + (p – 1)x + 1)
1
2005 Paper 2 Q11
(b) Hence find the range of values of p for which all the roots are real
7
From (a)
x3 + px2 + px + 1 = (x + 1)(1x2 + (p – 1)x + 1)
a=1
b = (p – 1)
c=1
If real then b2 – 4ac ≥ 0
(p –
– 4(1)(1) ≥ 0
p2 – 2p + 1 – 4 ≥ 0
p2 – 2p – 3 ≥ 0
(p – 3)(p + 1) ≥ 0
1)2
 Real when p ≤ -1 & p ≥ 3
y
-1
3
x
7
2006 Paper 1 Q8
(a) Express 2x2 + 4x – 3 in the form
f(x) = a(x + b)2 + c
(b) Write down the coordinaates of the turning point
3
1
2x2 + 4x – 3 = 2[x2 + 2x – 3/2]
= 2[(x2 + 2x + 1) – 1 – 3/2]
= 2[(x2 + 2x + 1) – 2/2 – 3/2]
= 2[(x + 1)2 – 5/2]
= 2(x + 1)2 – 5
Happy as + x2  (- 1, - 5) is Min Tpt
4
2007 Paper 1 Q4
Find the range of values of k such that the equation kx2 – x – 1 = 0 has no real roots
a=k
b = -1
c = -1
kx2 – 1x – 1= 0
b2 – 4ac < 0  No Real Roots
(-1)2 – 4(k)(-1) < 0
1 + 4k < 0
4k < – 1
k<–¼
4
2008 Paper 1 Q10 Which of the following are true about the equation x2 + x + 1 = 0
(1) The roots are equal
(2) The roots are real
A = Neither
B = (1) Only
C = (2) Only
D = Both are True
a=1
b=1
c=1
1x2 + 1x + 1= 0
Find nature of b2 – 4ac
(1)2 – 4(1)(1) = 1 – 4 = -3 < 0
As b2 – 4ac < 0  Non real roots are not equal or real
 A – Neither (1) or (2) are true
2
2008 Paper 1 Q13
The graph has an equation of the form
What is the equation of the graph
y = k(x – a)(x – b).
Use roots at x = 1 & x = 4 for factors
y = k(x – 1)(x – 4)
12 = k(0 – 1)(0 – 4)
12 = k(-1)(-4)
4k = 12
k=3

y = 3(x – 1)(x – 4)

A
y
12
1
4
x
2
2008 Paper 1 Q16
2x2 + 4x + 7 is expressed in the form
What is the value of q?
2(x + p)2 + q
2
2x2 + 4x + 7 = 2[x2 + 2x + 7/2]
= 2[(x2 + 2x + 1) – 1 + 7/2]
= 2[(x2 + 2x + 1) – 2/2 + 7/2]
= 2[(x + 1)2 + 5/2]
= 2(x + 1)2 + 5
= 2(x + p)2 + q
q=5=A
2
2006 Paper 1 Q8
(a) Write x2 – 10x + 27 in the form (x + b)2 + c
(b) Hence show that g(x) = 1/3 x3 – 5x2 + 27x – 2 is always increasing
(a)
2
4
x2 – 10x + 27 = (x2 – 10x + 25) – 25 + 27
= (x – 5)2 + 2
Increasing  Differentiate g(x) = 1/3 x3 – 5x2 + 27x – 2
(b)