Download 6 Roots, Surds and Discriminant

“Teach A Level Maths”
Vol. 1: AS Core Modules
6: Roots, Surds and
Discriminant
© Christine Crisp
Roots, Surds and Discriminant
Module C1
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Roots, Surds and Discriminant
Roots of Equations
Roots is just another word for solutions !
2
e.g. Find the roots of the equation x  2 x  1  0
Solution: There are no factors, so we can either
complete the square or use the quadratic formula.
2
Completing the square: ( x  1)  1  1  0


( x  1)   2
x  1  2
Using the formula:
 b  b  4ac
x
2a
2

x
2
(1)
( 2) 2  4(1)(1)
2
2 8
(2)
 x
2
Let’s see why the answers (1) and (2) are the same
Roots, Surds and Discriminant
The answers from the quadratic formula can be
simplified:
2 8
We have x 
2
Numbers such as 8 are called surds
However, 8  4  2
4 is a perfect square so can be square-rooted, so
4 2  2 2
We have simplified the surd
2 8
22 2

So, x 
x
2
2
2 is a common factor of the numerator, so
1
2(1  2 )
x
 x  1  2
21
Roots, Surds and Discriminant
Exercise
Simplify the following surds:
1)
12 
4 3  2 3
2)
32 
16  2  4 2
3)
45 
9 5  3 5
4) 1000 
100  10  10 10
Roots, Surds and Discriminant
The Discriminant of a Quadratic Function
The formula for solving a quadratic equation is
 b  b 2  4ac
x
2a
The part b  4ac is called the discriminant
2
Because we square root the discriminant, we get
different types of roots depending on its sign.
Roots, Surds and Discriminant
The Discriminant of a Quadratic Function
To investigate the roots of the equation
x 2  5x  4  0
we consider the graph of the function y  x 2  5 x  4
The roots of the
equation are at the
points where y = 0
( x = 1 and x = 4)
The discriminant
y  x2  5x  4
b 2  4ac  25  16
9 0
The roots are real and distinct.
( different )
Roots, Surds and Discriminant
The Discriminant of a Quadratic Function
For the equation x 2  4 x  4  0
the discriminant b 2  4ac  16  16  0
y  x2  4x  4
The roots are real and equal
( x = 2)
Roots, Surds and Discriminant
The Discriminant of a Quadratic Function
For the equation x 2  4 x  7  0
. . .
. . . the discriminant b 2  4ac  16  28
 12  0
y  x 2  4x  7
There are no real roots
as the function is never
equal to zero
If we try to solve x 2  4 x  7  0 , we get
4   12
x
2
The square of any real number is positive so there
are no real solutions to  12
Roots, Surds and Discriminant
SUMMARY
The formula for solving the quadratic equation
ax  bx  c  0
2
is
 b  b 2  4ac
x
2a
The part b 2  4ac is called the discriminant
b 2  4ac  0 The roots are real and distinct
b 2  4ac  0
( different )
The roots are real and equal
b 2  4ac  0 The roots are not real
If we try to solve an equation with no real roots, we will
be faced with the square root of a negative number!
Roots, Surds and Discriminant
Exercise
1 (a) Use the discriminant to determine the nature of
the roots of the following quadratic equations:
(i) x 2  2 x  1  0
(ii)
x  2x  1  0
2
(b) Check your answers by completing the square to
find the vertex of the function and sketching.
Solution: (a) (i)
b 2  4ac  4  4(1)(1)  0
The roots are real and equal.
(ii) b 2  4ac  4  4(1)( 1)  4  4  8
The roots are real and distinct.
Roots, Surds and Discriminant
(b) Check your answers by completing the square to
find the vertex of the function and sketching.
(b) (i)
x 2  2x  1
 (x  1) 2  1  1
 ( x  1) 2
y  x2  2x  1
 Vertex is ( -1,0 )
Roots of equation
(real and equal)
(ii)
x 2  2 x2  1
 (x  1)  1  1
 (x  1) 2  2
 Vertex is ( 1,-2 )
Roots of equation
(real and distinct)
y  x2  2x  1
Roots, Surds and Discriminant
2. Determine the nature of the roots of the
following quadratic equations ( real and distinct or
real and equal or not real ) by using the
discriminant. DON’T solve the equations.
(a)
x 2  6x  9  0
b 2  4ac  (6) 2  4(1)(9)  36  36  0
Roots are real and equal
(b)
2x 2  5x  9  0
b 2  4ac  (5) 2  4(2)(9)  25  72  47  0
There are no real roots
(c)
5x 2  9x  2  0
b 2  4ac  (9) 2  4(5)( 2)  81  40  121  0
Roots are real and distinct
Roots, Surds and Discriminant
Roots, Surds and Discriminant
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Roots, Surds and Discriminant
Roots is just another word for solutions !
2
x
 2x  1  0
e.g. Find the roots of the equation
Solution: There are no factors, so we can either
complete the square or use the quadratic formula.
2
Completing the square: ( x  1)  1  1  0


( x  1)   2
x  1  2
 b  b  4ac
x
2a
 2  2 2  4(1)( 1)
x
2
2 8
x
2
Using the formula:
2


(1)
(2)
Roots, Surds and Discriminant
The answers from the quadratic formula can be
simplified:
2 8
We have x 
2
Numbers such as 8 are called surds
However, 8  4  2
4 is a perfect square so can be square-rooted, so
4 2  2 2
We have simplified the surd
2 8
22 2
 x
So, x 
2
2
2 is a common factor of the numerator, so
1
2(1  2 )
x
 x  1  2
21
Roots, Surds and Discriminant
The Discriminant
The formula for solving the quadratic equation
2

b

b
 4ac
2
x
ax  bx  c  0 is
2a
The part b 2  4ac is called the discriminant
b 2  4ac  0
The roots are real and distinct.
( different )
The roots are real and equal.
b 2  4ac  0
The roots are not real.
b 2  4ac  0
If we try to solve an equation with no real roots,
we will be faced with the square root of a negative
number!
Roots, Surds and Discriminant
e.g. For
y  x2  5x  4
The discriminant
b 2  4ac  25  16
9
0
y  x2  5x  4
The roots are real and distinct.
( different )
Roots, Surds and Discriminant
e.g. For y  x 2  4 x  4
The discriminant
b 2  4ac  16  16
0
y  x2  4x  4
The roots are real and equal.
Roots, Surds and Discriminant
e.g. For y  x 2  4 x  7
The discriminant
b 2  4ac  16  28
 12
0
y  x2  4x  7
There are no real roots as the function is never
equal to zero.