Wilson’s School Core 1 Revision Sheet 2 of 5
Polynomials
Stuff to memorise:
 b  b 2  4ac
2a
2
2
The discriminant of ax  bx  c is b  4ac
The solution of ax 2  bx  c  0 is x 
b 2  4ac  0
b 2  4ac  0
b 2  4ac  0
If
The equation
ax 2  bx  c  0
has
2 distinct real roots
1 (repeated) real root
no real roots
I can:
Add, subtract and multiply polynomials
Complete the square for a quadratic expression
Use the discriminant to determine the number of real roots of a
quadratic equation
Solve quadratic equations by factorising, completing the square
and using the quadratic formulae
Solve linear and quadratic inequalities
Solve simultaneous equations where one equation is linear and the
other quadratic
Recognise and solve ‘hidden’ quadratic equations
Tick!
Test yourself:
1
2
3
4
5
6
7
8
9
10
11
Solve 6 x 2  x  15 , giving each answer as an exact fraction
Hence write down the solution set of the inequality 6 x 2  x  15
Find integers p and q such that x 2  8 x  11  ( x  p) 2  q
Hence solve x 2  8 x  11  0 giving your answers in surd form
4 16
2
c) And solve 2   11  0 (hint: let x  in part b)
x
a
x
2
Solve the inequality ( x  7)  4
Solve x 4  5 x 2  6  0 using the substitution a  x 2
Find the constants a , b and c such that 3x 2  5 x  1  a( x  b) 2  c .
Hence find the coordinates of the vertex.
Find the range of values of k for which the equation x 2  3kx  k  0 has
any real roots.
Solve the inequality x  x 3  0
3x  y  7
Solve the simultaneous equations 2
x  2 xy  5 y  11
a)
b)
a)
b)
a) Factorise 9  3x  2 x 2
b) Sketch the graph y  9  3x  2 x 2 showing intercepts
Given that ( x 2  ax  2)( x  b)  x 3  7 x 2  14 x  c , find the possible values
of a, b and c.
Expand and simplify the expression (3x  4)( x  1)( 2 x  3)
Solutions
1 a) 3/2 or -5/3
b) x  3 / 2 or x  5 / 3
2 a) p=4, q=-5
b)  4  5
3
4
c)
4 5
 9  x  5
6
5 a=3, b=-5/6, c=-13/12
 5 13 
vertex=  , 
 6 12 
4
9
7 -1<x<0 and x>1
 8 25 
8 (3,2)  , 
7 7 
9 a) (3-2x)(x+3)
b)
10 a=4, b=3, c=-6 or a=3, b=4 c=-8
11 6 x 3  5 x 2  13x  12
6
2
k  0 and k 

2(4  5 )
11
d) -5