Download Word

Further Concepts for Advanced Mathematics - FP1
Unit 1 Complex Numbers – Section1a Quadratic Equations1
Quadratic Equations
“Be able to solve any quadratic equation with real coefficients.”
You should already know that any quadratic equation can be written in the form
ax 2  bx  c  0 where a, b and c are real numbers and a  0 .
Here is a list of quadratic equations and their coefficients:
Equation
3x 2  2 x  7  0
5x 2  4  0
2 x  8x 2  0
although this can be rearranged to
8x 2  2 x  0
Coefficients
a  3, b  2, c  7
a  5, b  0, c  4
a  8, b  2, c  0
a  8, b  2, c  0
There are obviously far too many variations on the same thing to write a really
comprehensive list. Just remember, the x 2 term must be present and be the highest
power of x and that there should be no other powers of x than x 1 (i.e. x) and/or x 0 (i.e.
a number).
Solving Quadratic Equations
You should know about the methods used to solve quadratic equations from G.C.S.E.
They are:
1.
2.
3.
4.
By factorisation
By completing the square
By using the quadratic formula
By drawing a graph
Factorisation
These examples should help to remind you how factorisation can be used to solve
quadratic equations.
1.
x 2  2 x  15  0
( x  3)( x  5)  0
x  3 or x  5
2.
5x 2  2 x  0
x(5 x  2)  0
x  0 or x 
2
5
The numbers -3 and 5 are referred to as the roots of the
equation x 2  2 x  15  0 .
The roots of the equation 5 x 2  2 x  0 are 0 and
1
2
5
3.
4.
5.
6 x 2  x  12  0
(3x  4)( 2 x  3)  0
4
3
x  or x  
3
2
The roots of the equation 6 x 2  x  12  0 are
9x 2  4  0
(3x  2)(3x  2)  0
2
2
x   or x 
3
3
The roots of the equation 9 x 2  4  0 are 
4
3
and 
3
2
2
2
and
3
3
x2  5  0
( x  5 )( x  5 )  0
x 5
5 and  5
The roots of the equation x 2  5  0 are
The last two examples use the factorisation method known as “the difference of two
squares”.
Completing the Square
These examples should remind you how to use the method of completing the square
1.
x 2  4x  1  0
( x  2) 2  4  1  0
( x  2) 2  3  0
( x  2) 2  3
x2  3
x  2 3
2.
The roots of x 2  4 x  1  0 are 2  3 and 2  3
3x 2  2 x  5  0
x 2  23 x  53  0
dividing all terms by 3
( x  13 ) 2  19  53  0
( x  13 ) 2  169
x  13  
16
9
x  13  43
x
3.
5
3
The roots of 3 x 2  2 x  5  0 are
or x  1
2 x 2  8 x  11  0
x 2  4 x  112  0
5
3
and  1
dividing all terms by 2
( x  2)  4  112  0
2
( x  2) 2  192  0
( x  2) 2  192
x2
19
2
x  2 
19
2
The roots of 2 x 2  8 x  11  0 are  2 
2
19
2
and  2 
19
2
In examples 2 and 3 the coefficient of x 2 has been reduced to 1 by division. This is a
useful trick when solving quadratic equations.
The Quadratic Formula
 b  b 2  4ac
where a is the coefficient of x 2 , b is the
2a
coefficient of x and c is the coefficient of x 0 (i.e. the number at the end).
The quadratic formula is x 
These examples should remind you how the formula works:
1.
x 2  5x  3  0
a  1, b  5, c  3
5  25  12
2
5  23
x
2
5  23
5  23
x
or x 
2
2
x
2.
3x 2  2 x  5  0
a  3, b  2, c  5
 2  4  60
6
 2  64
x
6
28
x
6
5
x  1 or x  
3
x
This example could have been done by
factorising the original expression.
Drawing a Graph
This example should remind you how a graph can be used to solve a quadratic equation.
1.
x 2  3x  7  0
y  x 2  3x  7
The graph crosses the x axis at x = -4.5 and
x = 1.5 (both to 1 decimal place).
3
In all of the methods it is very important to start from the position where the equation is
written in the form ax 2  bx  c  0 (the = 0 makes life as easy as possible).
4