Download Complex solutions to quadratic equations.

Complex solutions to quadratic equations.
Complex solutions to quadratic equations.
1 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Complex solutions to quadratic equations.
2 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Nope, you have to consider rational numbers to solve this kind of equation.
Complex solutions to quadratic equations.
2 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Nope, you have to consider rational numbers to solve this kind of equation.
Is there a solution to x 2 = 2 if you are only allowed to look in the rational
numbers?
Complex solutions to quadratic equations.
2 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Nope, you have to consider rational numbers to solve this kind of equation.
Is there a solution to x 2 = 2 if you are only allowed to look in the rational
numbers?
Nope, you have to consider real numbers to solve this kind of equation.
Complex solutions to quadratic equations.
2 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Nope, you have to consider rational numbers to solve this kind of equation.
Is there a solution to x 2 = 2 if you are only allowed to look in the rational
numbers?
Nope, you have to consider real numbers to solve this kind of equation.
Is there a solution to x 2 = −1 if you are only allowed to look in the real
numbers?
Complex solutions to quadratic equations.
2 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Nope, you have to consider rational numbers to solve this kind of equation.
Is there a solution to x 2 = 2 if you are only allowed to look in the rational
numbers?
Nope, you have to consider real numbers to solve this kind of equation.
Is there a solution to x 2 = −1 if you are only allowed to look in the real
numbers?
Nope, you have to consider complex numbers to solve this kind of
equation.
Complex solutions to quadratic equations.
2 / 10
Complex numbers
Motivation:
Is there a solution to 2x = 1 if you are only allowed to look in the integers?
Nope, you have to consider rational numbers to solve this kind of equation.
Is there a solution to x 2 = 2 if you are only allowed to look in the rational
numbers?
Nope, you have to consider real numbers to solve this kind of equation.
Is there a solution to x 2 = −1 if you are only allowed to look in the real
numbers?
Nope, you have to consider complex numbers to solve this kind of
equation.
We will add to the reals a symbol which represents a square root of −1.
Complex solutions to quadratic equations.
2 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
Complex solutions to quadratic equations.
3 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
A complex number is a quantity of the form a + bi where a and b are
real numbers.
Complex solutions to quadratic equations.
3 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
A complex number is a quantity of the form a + bi where a and b are
real numbers.
If b = 0 then a + bi is called a real number, if a = 0 then a + bi is called
imaginary.
Complex solutions to quadratic equations.
3 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
A complex number is a quantity of the form a + bi where a and b are
real numbers.
If b = 0 then a + bi is called a real number, if a = 0 then a + bi is called
imaginary.
Can we do arithmetic with complex numbers?
Complex solutions to quadratic equations.
3 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
A complex number is a quantity of the form a + bi where a and b are
real numbers.
If b = 0 then a + bi is called a real number, if a = 0 then a + bi is called
imaginary.
Can we do arithmetic with complex numbers?
Addition and subtraction:
What should (a + bi) + (c + di) mean?
a + bi + c + di =
Complex solutions to quadratic equations.
3 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
A complex number is a quantity of the form a + bi where a and b are
real numbers.
If b = 0 then a + bi is called a real number, if a = 0 then a + bi is called
imaginary.
Can we do arithmetic with complex numbers?
Addition and subtraction:
What should (a + bi) + (c + di) mean?
a + bi + c + di = (a + c) + (b + d)i
In order to add / subtract complex numbers just add / subtract the real
and imaginary parts.
Complex solutions to quadratic equations.
3 / 10
Complex numbers
The imaginary unit i is a formal symbol which we define to have the
property that i2 = −1.
A complex number is a quantity of the form a + bi where a and b are
real numbers.
If b = 0 then a + bi is called a real number, if a = 0 then a + bi is called
imaginary.
Can we do arithmetic with complex numbers?
Addition and subtraction:
What should (a + bi) + (c + di) mean?
a + bi + c + di = (a + c) + (b + d)i
In order to add / subtract complex numbers just add / subtract the real
and imaginary parts.
Compute (2 + 3i) − (3 + 2i).
Complex solutions to quadratic equations.
3 / 10
multiplication
The FOIL’ing rule lets us multiply complex numbers:
(a + bi) · (c + di) =
Complex solutions to quadratic equations.
4 / 10
multiplication
The FOIL’ing rule lets us multiply complex numbers:
(a + bi) · (c + di) = ac + adi + bci + bdi2 =
Complex solutions to quadratic equations.
4 / 10
multiplication
The FOIL’ing rule lets us multiply complex numbers:
(a + bi) · (c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i
Complex solutions to quadratic equations.
4 / 10
multiplication
The FOIL’ing rule lets us multiply complex numbers:
(a + bi) · (c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i
Compute:
(2 + 3i) · (2 − 3i)
(1 + i)4
Complex solutions to quadratic equations.
4 / 10
Complex conjugation
For a complex number z = a + bi, its conjugate is given by
z = a + bi =
Complex solutions to quadratic equations.
5 / 10
Complex conjugation
For a complex number z = a + bi, its conjugate is given by
z = a + bi = a − bi
To compute the conjugate just negate the imaginary part.
Notice that
z · z = (a + bi) · (a + bi) =
Complex solutions to quadratic equations.
5 / 10
Complex conjugation
For a complex number z = a + bi, its conjugate is given by
z = a + bi = a − bi
To compute the conjugate just negate the imaginary part.
Notice that
z · z = (a + bi) · (a + bi) = (a + bi)(a − bi) =
Complex solutions to quadratic equations.
5 / 10
Complex conjugation
For a complex number z = a + bi, its conjugate is given by
z = a + bi = a − bi
To compute the conjugate just negate the imaginary part.
Notice that
z · z = (a + bi) · (a + bi) = (a + bi)(a − bi) = ... =
Complex solutions to quadratic equations.
5 / 10
Complex conjugation
For a complex number z = a + bi, its conjugate is given by
z = a + bi = a − bi
To compute the conjugate just negate the imaginary part.
Notice that
z · z = (a + bi) · (a + bi) = (a + bi)(a − bi) = ... = a2 + b 2 + 0i
Complex solutions to quadratic equations.
5 / 10
Complex conjugation
For a complex number z = a + bi, its conjugate is given by
z = a + bi = a − bi
To compute the conjugate just negate the imaginary part.
Notice that
z · z = (a + bi) · (a + bi) = (a + bi)(a − bi) = ... = a2 + b 2 + 0i
Multiplying a complex number by its conjugate always produces a positive
real number.
Compute (2 + 3i) · (2 + 3i).
Complex solutions to quadratic equations.
5 / 10
Dividing a complex number by a real number
To divide a complex number by a real number just divide the real and
imaginary components:
a + bi
=
c
Complex solutions to quadratic equations.
6 / 10
Dividing a complex number by a real number
To divide a complex number by a real number just divide the real and
imaginary components:
a + bi
a b
= + i
c
c
c
It is harder to divide by a complex number.
a + bi
How can we make sense of
?
c + di
Multiply and divide by the conjugate!
a + bi
=
c + di
Complex solutions to quadratic equations.
6 / 10
Dividing a complex number by a real number
To divide a complex number by a real number just divide the real and
imaginary components:
a + bi
a b
= + i
c
c
c
It is harder to divide by a complex number.
a + bi
How can we make sense of
?
c + di
Multiply and divide by the conjugate!
a + bi
(a + bi)(c − di)
=
=
c + di
(c + di)(c − di)
Complex solutions to quadratic equations.
6 / 10
Dividing a complex number by a real number
To divide a complex number by a real number just divide the real and
imaginary components:
a + bi
a b
= + i
c
c
c
It is harder to divide by a complex number.
a + bi
How can we make sense of
?
c + di
Multiply and divide by the conjugate!
a + bi
(a + bi)(c − di)
ac + bd + (bc − ad)i
=
=
=
c + di
(c + di)(c − di)
c2 + d2
Complex solutions to quadratic equations.
6 / 10
Dividing a complex number by a real number
To divide a complex number by a real number just divide the real and
imaginary components:
a + bi
a b
= + i
c
c
c
It is harder to divide by a complex number.
a + bi
How can we make sense of
?
c + di
Multiply and divide by the conjugate!
a + bi
(a + bi)(c − di)
ac + bd + (bc − ad)i
=
=
=
c + di
(c + di)(c − di)
c2 + d2
ac + bd
bc − ad
+ 2
i
2
2
c +d
c + d2
1+i
2+i
Compute
and
2+i
1 + 2i
Complex solutions to quadratic equations.
6 / 10
Square roots of negative numbers
We’ve formally added a solution to x 2 = −1. Have we added more than
that?
√
Compute (i 7)2 =
Complex solutions to quadratic equations.
7 / 10
Square roots of negative numbers
We’ve formally added a solution to x 2 = −1. Have we added more than
that?
√
√
Compute (i 7)2 = i2 · ( 7)2 =
Complex solutions to quadratic equations.
7 / 10
Square roots of negative numbers
We’ve formally added a solution to x 2 = −1. Have we added more than
that?
√
√
Compute (i 7)2 = i2 · ( 7)2 = − 7
√
√
We have also added a square root to −7: −7 = i 7
The
√ same√is true for all negative numbers,
−n = i n.
Complex solutions to quadratic equations.
7 / 10
Solutions to quadratics
Solve x 2 + 3x + 5 = 0:
By the quadratic formula the solution is
x=
Complex solutions to quadratic equations.
8 / 10
Solutions to quadratics
Solve x 2 + 3x + 5 = 0:
By the quadratic formula the solution is
√
−3 ± 32 − 4 · 5
x=
=
2
Complex solutions to quadratic equations.
8 / 10
Solutions to quadratics
Solve x 2 + 3x + 5 = 0:
By the quadratic formula the solution is
√
√
−3 ± 32 − 4 · 5
−3 ± −11
x=
=
=
2
2
Complex solutions to quadratic equations.
8 / 10
Solutions to quadratics
Solve x 2 + 3x + 5 = 0:
By the quadratic formula the solution is
√
√
√
−3 ± 32 − 4 · 5
−3 ± −11
−3 ± i 11
x=
=
=
2
2
2
Complex solutions to quadratic equations.
8 / 10
Solutions to quadratics
Solve x 2 + 3x + 5 = 0:
By the quadratic formula the solution is
√
√
√
−3 ± 32 − 4 · 5
−3 ± −11
−3 ± i 11
x=
=
=
2
2
2
Quadratic equations always have solutions, provided you allow complex
numbers to appear.
For you: Solve x 2 − 2x + 2 = 0.
Complex solutions to quadratic equations.
8 / 10
How are complex roots related to each other?
So far we have seen that:
The solutions
to x 2 + 3x√+ 5 = 0 are
√
11
−3
11
−3
+
i and
−
i
2
2
2
2
The solutions to x 2 − 2x + 2 = 0 are
1 + i and 1 − i
They are related by complex conjugation.
Theorem
If the quadratic equation ax 2 + bx + c = 0 has a complex solution a + bi
then its other solution is the complex conjugate: a − bi
Complex solutions to quadratic equations.
9 / 10
Some practice
Solve
1
x2 + 4 = 0
2
x2 − 4 = 0
3
x 2 + 4x + 9 = 0
4
x 2 − 4x − 9 = 0
5
x 2 + 4x + 4 = 0
6
x 2 − 4x − 4 = 0
Complex solutions to quadratic equations.
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