Impromptu Problems
3
1.
Write the expression √𝑥 ÷ √𝑥 using just a single radical.
2.
Multiply out each of the following expressions.
a) 4𝑥 3 (3𝑥 4 − 2𝑥 2 − 5𝑥 + 2)
3.
Simplify each of the following expressions.
a)
4.
b) (𝑥 2 + 𝑥 + 4)(𝑥 + 3)
1
𝑥 2 −5𝑥+8−32
−1
b) 𝑥+1𝑥
𝑥+3
Find the exact solution of each of the following.
a) (𝑥 2 − 13)3 = 27
b) √𝑥 + 1 = 𝑥 − 1
5.
Solve each of the following for x. Find the exact value.
a) 4𝑥 = 13
b) log(𝑥) + log(𝑥 − 3) = 1
6.
Which number is larger the number of inches in a mile or the number of Fridays that will occur in the
next 1000 years? Explain how you obtained your answer?
7.
Find the exact solution to each of the following trigonometric equations.
−1
a) cos(5𝜃) = 2
b) 2sin2 (𝜃) − sin 𝜃 = 0
8.
Two friends who live 33 miles apart ride bikes toward each other. One averages 12 mph, and the other
averages 10 mph. How long will it take for them to meet?
9.
Solve the inequality 1 ≤ −7𝑥 + 8 ≤ 15. Express your answer as an inequality and graph it on a number
line.
10.
If the slope of a line is −1
and the line passes through the point (1, 5), express the equation of the line
3
without using any fractions.
11.
Graph each of the following equations finding both their x and y intercepts.
a) 3𝑥 − 4𝑦 = 12
b) 𝑦 − 2 = 12(𝑥 − 4)
12.
If 3 times the square of a positive number is added to 5 times the number the result is 2. Find the
number.
13.
Solve for 𝐵: 𝐻 = 𝑅+𝐵
14.
The sum of two numbers is -18. One number is 2 more than 3 times the other. Find the product of the
two numbers.
15.
Factor completely. 24𝑚3 − 14𝑚2 − 24𝑚
16.
Simplify each of the following.
a) √32𝑥 + √2𝑥 − √18𝑥
2
𝑅𝐵
b) (√𝑥 − √𝑦)(√𝑥 + √𝑦)
14
17.
Rationalize the denominator of
18.
If the length of each side of a square is decreased by 4 inches, the perimeter of the new square is 10
inches more than half the perimeter of the original square. What are the dimensions of the original
square?
19.
Find the exact solution to each equation.
a) 𝑥 2 − √5𝑥 − 1 = 0
√11−√7
and simplify.
b) (𝑥 + 4)(𝑥 + 2) = 2𝑥
20.
Solve the inequality 2𝑥 3 − 3𝑥 2 − 5𝑥 < 0
21.
If 𝑓(𝑥) = 𝑥 2 − 5𝑥 + 3 find and simplify
22.
If the graph of 𝑦 = 𝑓(𝑥) is shown to the right. Find the
graph of 𝑦 = −2𝑓(𝑥 + 2) − 3.
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
3
2
1
3
2
1
1
2
3
1
2
3
23.
Find a polynomial of degree 3 with -2, 1, and 4 as zeros, and 𝑓(2) = 16.
24.
Find the exact solution to the equation 6𝑥−3 = 34𝑥+1
25.
Graph the equation 𝑥 2 + 𝑦 2 − 12𝑥 + 10𝑦 = −25, give the name of the graph.
26.
Find the equation of the perpendicular bisector of the line segment whose endpoints are the points
(4, −5) and (10,3).
27.
In a particular teachers’ union, a teacher may retire when the teacher’s age plus the teacher’s years of
service is at least 90. If a 37 year old teacher has been teaching for 15 years, at what age will they be
eligible to retire?
28.
Find all points on the y-axis that are a distance 13 from the point (12,6)?
29.
Find the equation of the circle that has its center at the point (−5, −1) and is tangent to the line 𝑥 = 4.
30.
Express the function 𝑓(𝑥) = −2𝑥 2 + 12𝑥 − 14 in the form 𝑎(𝑥 − ℎ)2 + 𝑘.
31.
The American with Disabilities Act of 1990 guarantees all persons the right of accessibility of public
accommodations. Providing access to a building often involves a wheelchair ramp. Ramps should have 1
inch of vertical rise for every 12 – 20 inches of horizontal run. If the base of a doo is located 3 feet
above a sidewalk, determine the range of appropriate lengths for a wheelchair ramp.
32.
Write an expression for 𝑔(𝑥) if the graph of 𝑔 is obtained from the graph of 𝑓(𝑥) = 12𝑥 − 3 by reflecting
𝑓 about the (a) x-axis, (b) y-axis, (c) line 𝑦 = 2, (d) line 𝑥 = 3.
𝑥
33.
Without using a graphing calculator graph 𝑓(𝑥) = (𝑥+5)(𝑥 2 −5𝑥+4).
34.
If a square with sides parallel to the coordinate axes is inscribed in the ellipse 𝑎𝑥2 + 𝑦𝑏2 = 1, with 𝑎 > 𝑏,
express the area of the square in terms of a and b.
35.
Express sin(7𝑡) sin(4𝑡) as a sum or difference of sin and cos.
36.
Find the exact value of sec (arcsin(32)).
37.
Verify the trigonometric identity: sin 𝜃+cos 𝜃 = csc 𝜃 − cot 𝜃
38.
Find the distance between the parallel lines 4𝑥 + 3𝑦 + 12 = 0 and 4𝑥 + 3𝑦 − 38 = 0.
39.
Find the equation of the line which is tangent to the circle 𝑥 2 + 𝑦 2 + 8𝑥 + 6𝑦 + 8 = 0 at the point
(−8, −2).
40.
In the figure to the right, lines l and m are
parallel. What is the value for y?
2
2
cot 𝜃−tan 𝜃
(y+50)º
(4y-7)º
l
m