Complex Numbers and Operations
What is the difference of (6 + 4i) and (1 + 8i)?
(6 + 4i) - (1 + 8i)
(6 + 4i) + (-1 - 8i)
(6 - 1) + (4i - 8i)
5 - 4i
Solve for x by completing the square:
x2 - 18x + 81 = 4
x2 - 18x + 81 = 4
(x - 9)2 = 4
x - 9 = -2, 2
x = 7, 11
Describe the nature of the roots for the equation
7x2 - 12x + 1 = 0
(Hint: Use the discriminant b2 - 4ac)
b2 - 4ac
(-12)2 - 4(7)(1)
144 - 28 > 0
The equation has two real roots.
Use square roots to solve the equation x2 = -100 over the complex numbers.
x2 = -100
x = -10i, 10i
Calculate the value that completes the square.
x2 + 20x + 35 = 0
(20/2)2
(10)2
100
Solve x2 + 6x - 8 = 0 using the Quadratic Formula.
x = -3 ±√17
Factor the sum of squares.
16x2 + 9
16x2 + 9
(4x + 3i)(4x - 3i)
Write the equation x2 + 20x + 35 = 0 in the form
(x + p)2 = q
x2 + 20x + 35 = 0
x2 + 20x + 100 = -35 + 100
(x + 10)2 = 65
Solve 3x2 + 2x - 5 = 0 using the Quadratic Formula.
x = 1, -5/3
Write the product (1 + 2i)(1 - 2i) in the form a + bi.
(1 + 2i)(1 - 2i)
1 - 2i + 2i -4i2
1 - 4(-1)
1 + 4
5
Solve 0 = x2 + 14x + 4 by completing the square.
0 = x2 + 14x + 4
-4 = x2 + 14x
-4 + 49 = x2 + 14x + 49
45 = (x + 7)2
±√45 = x + 7
±√(9*5) = x + 7
-7 ± 3√5 = x
Ms. K tosses a ball into the air. The function
h(t) = -5t2 + 8t + 16
gives the approximate height, h, in meters, of the ball t seconds after she tosses it. Does the ball reach a height of 15 m?
The ball will reach this height if a there are real roots to the equation when it is equal to 15.
-5t2 + 8t + 16 = 15
-5t2 + 8t + 1 = 0
[ b2 - 4ac]
82 - 4(-5)(1)
64 + 20 > 0
There are 2 real roots to this equation when it is equal to 15, therefore there are 2 real times at which the ball will reach a height of 15 m.
Write the quotient 17 / (4 - i) in the form a + bi.
4 + i