1. The roots of the quadratic equation z^2 + az + b = 0 are 2 - 3i and 2 + 3i. What is a+b?
2. Find all pairs of real numbers (x,y) such that x + y = 6 and x^2 + y^2 = 28. If you find more than one pair, then list your pairs in order by increasing x value, separated by commas. For example, to enter the solutions (2,4) and (-3,9), you would enter "(-3,9),(2,4)" (without the quotation marks).
3. For what real values of a does the quadratic 4x^2 + ax + 25 have nonreal roots? Give your answer as an interval.
4.Find one pair of real numbers, (x,y), such that x + y = 6 and x^3 + y^3 = 144.
5. For what real values of c is x^2 + 16x + c the square of a binomial? If you find more than one, then list your values separated by commas
6. For what real number k does the product (25 + ki)(3+2i) equal a real number?
7. What constant must we place in the blank below in order to be able to factor the resulting expression into the product of two linear factors:
2mn - 16m - 7n +__
8. What number can we place in the blank to make the quadratic below the square of a binomial:
x^2 + (5/3)x +__
What are you stuck on?
I would rewrite the first one as
1 z^2 + b z + c = 0
z = [-b +/- sqrt(b^2-4c)]/2
[z+{b -sqrt(b^2-4c)}][z+{b +sqrt(b^2-4c)}]=0
so
-b/2=2 so b = -4
.5 sqrt(16-4c) = 3i
sqrt (16-4c) = 6i
16-4c = -36
4c = 52
c = 13
z^2 -4z + 13
Well I got 9 for the first one, couldn't figure out the 2nd one, {6,0} for the 3rd one, (1,3) for the fourth one, -9,9 for the fifth one, 4 for the 6th one, 3 for the 7th one, and 16 for the last one.
I don't think I do too well when it gets to 2 variables.
I like this assignment. Of course, as Damon hinted at, we are not going to do this for you.
The questions require a basic understanding of these topics.
Where are your difficulties ?
Some hints:
#1 , see Damon's solution
or, alternate way:
sum of roots = 2 - 3i + 2 + 3i = 4
product of roots = (2 - 3i)(2 + 3i) = 4 - 9i^2 = 13
in z^2 + az + b = 0
sum of roots = -a/1 , so a = -4
product of roots = c/1, so c = 13
a+b = -4+13 = 9
#2 , y = 6-x
plug into the circle equation, and solve as a quadratic , hint:use the formula
#3 properties of the discriminant
#4, neat question, here is a short cut
x+y = 6
cube it
(x+y)^3 = 216
x^3 + 3x^2 y + 3x y^2 + y^3 = 216
x^3+y^3 + 3xy(x+y) = 216
144 + 3xy(6) = 216
18xy = 72
xy = 4
x(6-x) = 4
x^2 - 6x = -4
complete the square, instead of the formula
x^2 - 6x + 9 = -4+9
(x-3)^2 = 5
x-3 = +-SQRT(5)
x = 3 +-SQRT(5)
plug into y = 6-x to find the matching y values
#5 properties of perfect squares, I just used them in #4 when I completed the squrare
#6. expand it to get a quadratic in terms of k.
Set the discriminant equal to zero and solve for k
#7 2mn - 16m - 7n +__
= 2m(n - 8) - 7(n + ????)
#8 same type as #5