5.6 Quadratic Equations and Complex N...

5.6 Quadratic Equations and Complex Numbers
(by: nate epstine)

Discriminant= (b^2-4ac)- used to determine the number of REAL, UNIQUE solutions a quadratic equation has

If (b^2-4ac)>0, there are 2 distinct, real solutions
If (b^2-4ac)=0, there is 1 real solution (a double root)
If (b^2-4ac)<0, there are 0 real solutions


Example 1:
    Find the Discriminant of each equation, then determine the number of real solutions for each equation

A)    2x^2+4x+1=0        B)    2x^2+4x+2=0        C) 2x^2+4x+3=0
        4^2-4(2)(1)                    4^2-4(2)(2)                   4^2-4(2)(3)
               =8                               =0                                =(-8)  


Imaginary Number

Imaginary Number= i is defined to be the square root of -1

i^1    i^2    i^3    i^4
---------------------------
i       -1       -i       1

Example 4->    Use the Quadratic Formula to solve
3x^2-7x+5=0
x=(7±square root of (-7^2-4(3)(5))/2(3)
7±square root of (-11)/6


QUIZ THURSDAY→ 5.4-5.6
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Addition to Nate's Notes:

Complex Numbers: Any number that can be written as a+bi
a - real part
b - imaginary part


Notes Continued:(by: Halina A. Wiktor)

Division:  We don't really divide, we multiply by the Complex Conjugate. This is called factoring the denominator.

Complex Conjugate:  The complex conjugate of a +bi is denoted a+bi, is a-bi. All you do to find the conjugate is change the operator between the real and imaginary parts.

Ex. 9:  Find -2-3i  ->  -2+3i

Ex. 10:  Simplify 2+5i/2-3i   (Write your answer in standard form)
Step 1:  multiply the numerator and the denominator by the complex conjugate of the denominator  (2+5i)/(2-3i)  (2+3i)/(2+3i)
Step 2:  Foil and simplify    4+6i+10+15i^2/4+6+6i-9i^2 = -11+16i/13
Step 3:  Write the answer in standard form:   -11/13  +  16i/13

Graphing Complex Numbers:  Complex numbers are graphed in the complex plane.
Complex Plane:  The x-axis is the real axis and the y-axis is the imaginary axis.

Ex. 11: Graph 2-3 in the complex plane.

• Go over 1 on the real-axis and down 3 on the imaginary axis
• Put a dot there
• Draw a dotted line from the origin to that dot. (This is how i want complex numbers graphed)

Magnitude of a Complex Number:  denoted by |a+bi|, means: find the distance the complex is from the origin. Absolute value, means distance from zero. To do this:
|a+bi| = (square root of) a^2 +b^2

Ex. 12:  Evaluate |-2-3i| then sketch.
|-2-3i| = (square root of) (-2)^2 + (-3)^2 = (square root of) 4+9 = (square root of) 13

Ex. 13:  Which value is greater |2+4i|or |1-5i|? Prove algebraically
|2+4i|= (square root of) 2^2+ 4^2 + (square root of) 4+16 = (square root of) 20
|1-5| = (square root of) 1+25 = (square root of) 26 -> greater