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LC Math 1 Adv – Solving Quadratic Equations with Complex Solutions (LT’s 1, 2 &3)
In this lesson, we’ll be working with a new number system: the complex number system. To put things in
perspective, let’s point out that number systems are defined by humans for human purposes. The following
bit of history illustrates how and why humans define number systems.
Once upon a time, mathematicians reasoned that equations like x +5= 3 did not have solutions. These
mathematicians had not yet defined a system including negative numbers. Defining the negative numbers
allowed for equations like x +5= 3 to have solutions. Once mathematicians had defined negative numbers,
(
)
they also needed to develop rules for operations +,-,´,¸ involving negative numbers. This, generally, is
how number systems come to be defined and used.
In our last unit, we encountered a quandary similar to the one that negative numbers presented for early
mathematicians. More specifically, we encountered quadratic equations whose solving required taking
square roots of negative numbers. We concluded that these equations have no real solutions. Saying that
these equations have no real solutions implies that they possibly have some other type of solutions (nonreal???). The number system that mathematicians defined in order make these equations solvable is called
the complex number system, and it consists of the complex numbers. Our current unit will focus on
defining the complex numbers and developing rules for operations involving complex numbers.
The Basics: The complex numbers are founded on the definition of the imaginary number, i,
where i º -1 . Let’s take a quick “i test” to make sure we’re clear on the definition of the
imaginary number, i.
I. i 2 =
;
Justification:
II. i 3 =
;
Justification:
III. i 4 =
;
Justification:
IV. i5 =
;
Justification:
V. i 20 =
;
Justification:
VI. i 34 =
Justification:
VII. i55 =
Justification:
;
VIII. i 97 =
Justification:
;
;
A complex number is defined as a number that can be written in the form a+ bi , where a and b are real
numbers. a is the real part of the complex number, and bi is the imaginary part. For example, -3-5i is a
complex number with real part, -3 and imaginary part, -5i . (Think and Discuss: Is 0 a complex number?)
Solving Quadratic Equations with Complex Roots.
Solve each equation. Leave answers in simplest radical form.
1. x 2 = -5
2. -6x 2 = 120
3. 3x 2 +68 = 20
4. 8x 2 = -2
1
8
5. - x = 10
2
6. 12- x 2 = 27
Solve each equation by completing the square. Leave answers in simplest radical form.
1. x 2 + 4x = -5
2. -3x 2 -5x = 12
3. 3x 2 - 4x +8 = 0
4. 8x 2 + x = 3
5. x 2 = 2x -3
6. 25+ x 2 = 11x
Solve each equation using the quadratic formula. Leave answers in simplest radical form.
7. 2x 2 + 4x = -7
8. -4x 2 -3x = 9
9. 5x 2 -7x +6 = 0
10. 3x 2 +5x = 8
11. x 2 = 4x -5
12. 7+ x 2 = 8x
Extension: Write the standard form of a quadratic function with real coefficients that has roots 2±5i
and whose graph passes through (3 , 13). Show verification that your function satisfies all of the
specifications.