6: roots, surds and discriminant © christine crisp “teach a level maths” vol. 1: as core...
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6: Roots, Surds and 6: Roots, Surds and DiscriminantDiscriminant
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Roots, Surds and Discriminant
Module C1
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Roots, Surds and DiscriminantRoots of
Equations
e.g. Find the roots of the equation
0122 xx
Completing the square:
a
acbbx
2
42
0111 2 )(x
21 )(x 21 x
Using the formula:
Let’s see why the answers (1) and (2) are the same
2
82 x
)(1
)(2
2
)1)(1(4)2(2 2 x
Roots is just another word for solutions !Solution: There are no factors, so we can either complete the square or use the quadratic formula.
Roots, Surds and Discriminant
2
212 )( x
The answers from the quadratic formula can be simplified:
2
82 xWe
have
248 However,
2224
4 is a perfect square so can be square-rooted, so
2
82 xSo,
2
222 x
2 is a common factor of the numerator, so
21 x
Numbers such as are called surds
8
We have simplified the surd
1
1
Roots, Surds and Discriminant
Exercise
3234
101010100
Simplify the following surds:
12)1
32)2
45)3
1000)4
24216
5359
Roots, Surds and Discriminant
Because we square root the discriminant, we get different types of roots depending on its sign.
The Discriminant of a Quadratic FunctionThe formula for solving a quadratic equation is
a
acbbx
2
42
The part is called the discriminant
acb 42
Roots, Surds and Discriminant
452 xxy
we consider the graph of the function 452 xx
To investigate the roots of the equation
452 xx 0
yy0
The roots of the equation are at the points where y = 0
The roots are real and distinct.( different )
1625 The discriminantacb 42
9 0
The Discriminant of a Quadratic Function
( x = 1 and x = 4)
Roots, Surds and Discriminant
For the equation
0442 xxthe discriminant
acb 42 1616
442 xxy
The roots are real and equal
0
The Discriminant of a Quadratic Function
( x = 2)
Roots, Surds and Discriminant
For the equation . . . 0742 xx. . . the discriminant
acb 42 12
There are no real roots as the function is never equal to zero
2816
The Discriminant of a Quadratic Function
If we try to solve , we get0742 xx
2
124 x
The square of any real number is positive so there are no real solutions to 12
742 xxy0
Roots, Surds and Discriminant
042 acb
042 acb
The part is called the discriminant
acb 42
042 acb The roots are real and equalThe roots are not real
The roots are real and distinct( different )
SUMMARY
The formula for solving the quadratic equation
a
acbbx
2
42 02 cbxax is
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
1 (a) Use the discriminant to determine the nature of the roots of the following quadratic equations:
0114442 ))((acb
0122 xx(i)
(ii)
0122 xx(b) Check your answers by completing the
square to find the vertex of the function and sketching.
Solution: (a) (i) The roots are real and
equal.(ii) 844114442 ))((acb
The roots are real and distinct.
Exercise
Roots, Surds and Discriminant
122 xxy
122 xxy
(b) Check your answers by completing the square to find the vertex of the function and sketching.(b)
(i)
122 xx111 2 )(x
21)( x Vertex is ( -1,0 )
(ii)
122 xx111 2 )(x
21 2 )(x
Roots of equation(real and equal)
Roots of equation(real and distinct)
Vertex is ( 1,-2 )
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) 0962 xx
(c) 0295 2 xx
(b) 0952 2 xx
363691464 22 ))(()(acbRoots are real and equal
121408125494 22 ))(()(acb
47722592454 22 ))(()(acbThere are no real roots
Roots are real and distinct
0
0
0
Roots, Surds and Discriminant
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
Roots, Surds and Discriminant
Roots is just another word for solutions !e.g. Find the roots of the equation
0122 xx
Completing the square:
2
11422 2 ))(( x
a
acbbx
2
42
Solution: There are no factors, so we can either complete the square or use the quadratic formula. 0111 2 )(x
21 )(x 21 x
Using the formula:
2
82 x
(1)
(2)
Roots, Surds and Discriminant
2
212 )( x
The answers from the quadratic formula can be simplified:
2
82 xWe
have
248 However,
2224
4 is a perfect square so can be square-rooted, so
2
82 xSo,
2
222 x
2 is a common factor of the numerator, so
21 x
Numbers such as are called surds
8
We have simplified the surd
1
1
Roots, Surds and Discriminant
042 acb
042 acb
042 acb
The part is called the discriminantacb 42
The roots are real and equal.
The roots are not real.
The roots are real and distinct.( different )
The Discriminant
The formula for solving the quadratic equation
a
acbbx
2
42 02 cbxax is
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
452 xxy
The roots are real and distinct.( different )
1625 The discriminant
acb 42 90
e.g. For
452 xxy
Roots, Surds and Discriminant
acb 42 1616 0
The discriminant
e.g. For 442 xxy
442 xxy
The roots are real and equal.
Roots, Surds and Discriminant
742 xxy
acb 42 12
There are no real roots as the function is never equal to zero.
2816 0
The discriminant
e.g. For 742 xxy