# Math Test #1 Solutions

Solve the following equations:

(a) (3 + 2x)(1 - 3x) = (6x - 1)(1 - x),

(b) x(10x - 11) = 6

**Solutions:** (a) Expand both sides of the equation to
obtain:

(3+2x)(1-3x) = (6x-1)(1-x) 3-9x+2x-6x^{2} = 6x-6x^{2}-1+x

or

and so the solution written in set notation is

(b) Using the distributive property again we have

x(10x - 11) = 6 10x^{2} - 11x = 6
10x^{2} - 11x - 6 = 0:

Since the factorization method is not easy to apply here and also

the method of completing the square is still not very easy we apply

the quadratic formula a = 10, b = -11, c = -6:

or Observe that
and

which means the
relations are satisfied.

(c) Eliminating the denominators we have

then after checking we see the only solution is

Find all real solutions of the radical equations:

**Solutions:** (a) Squaring both sides, after isolating the
radical ex-

pression, we get just an implication but good enough to see how the

solutions might look like:

or x = 30 and after checking we see that

(b) Using the same method we get

32x^{2} - x - 3 = 0.

Using the quadratic formula

of which gives only one solution

(a) A motorboat maintained a constant speed of 18 miles
per hour

relative to the water in going 8 miles upstream and then returning.

The total time for the trip was 1 hours. Use this information to find

the speed of the current.

(b) If instead of knowing that it took 1 hours for the round trip

we know that it took 20 minutes more to go upstream than to go

downstream what would be the speed of the current in this case?

**Solutions: **(a) If we denote by x the speed of current in miles per

hour we get that the time going upstream is
(hours) and the

time going downstream is (hours). Hence
because it takes 1

hour round trip we obtain the equation:

Solving this for x by adding to the same common
denominator we

get:

which gives .

(b) Since 20 minutes is 1/3 of an hour. In this case the equation

becomes

which implies

This gives only the solution .

Solve the following inequalities and write your answers in in-

terval notation:

**Solutions:** (a) The inequality is equivalent to

which means

(b) The inequality becomes

(c) In this case we have similarly

(d) Since we are dealing with positive numbers the
inequality is

equivalent to

which gives

(a) Find the center and the radius of the circle of
equation

x^{2} + y^{2} + 39x - 80y = 0.

(b) What are the coordinates of the x-intercepts ?

**Solutions:** (a) The equation can be written after completing the

squares

or

which gives the center
and radius

(b) Making y = 0 in x^{2}+y^{2}+39x-80y = 0 we obtain x^{2}+39x = 0

which leads to two points of intersection:
and

Similarly for the y-intercept we get: and
. The graph

of this circle is included below:

Find the equation of the line in slope intercept form
containing

the point of coordinates (3,-2) and parallel to the line of equation

7x + 5y - 3 = 0.

**Solutions:** The slope of the given line is m =
. Hence the

equation of the line we are looking for is y - (-2) = (-7/5)(x - 3)

or .

Find the domain of the following functions

**Solutions:** (a) We need to impose the condition:

Hence

(b) In this case we have:

1. 2x - 1 ≥ 0 or and

2. or . If x
≥ 0, this is equivalent

to 2x - 1 = x^{2} or 0 = x^{2} - 2x + 1 (completing the square)

(x - 1)^{2} = 0. We have only one non-negative solution: x = 1.

Therefore

THE END