See also the
Dr. Math FAQ:
order of operations
Browse High School Basic Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Solving simple linear equations.
Positive/negative integer rules.
Completing the square.
Direct and indirect variation.
Inequalities and negative numbers.
- A Variable and Its Reciprocal, Each Cubed [11/20/2010]
Given a variable plus its multiplicative inverse, a student wonders how to evaluate a
sum of the same form, but where each of the unknowns has been cubed. Doctor Ali
suggests starting with a simpler problem before going on to show a few steps of
algebraic expansion and re-arrangement.
- Velocity and Time [02/11/1999]
A car averages 30 km/hr the first half of a trip, and 50 km/hr the second
half of the trip. What is the car's average velocity for the trip?
- A Velocity-Time Problem [02/16/1999]
If you drive 40 mph instead of 30 mph, you save 30 seconds per mile, but
if you make the same 10 mph increase from 70 mph to 80 mph, you only save
5.6 seconds per mile. Why?
- Vertex of a Parabola [04/16/2002]
Finding the y-coordinate of the vertex of a parabola.
- Vertical and Horizontal Asymptotes [6/3/1996]
What are the vertical and horizontal asymptotes?
- Visualizing 1 + 1/x [10/10/2003]
Show that the sum of a positive number and its reciprocal is at least 2.
- Walking the Shortest Distance [10/07/2002]
A single boy lives in each of n equally spaced houses on a straight
line. At what point should the boys meet so that the sum of the
distances that they walk from their houses is as small as possible?
- Water Level in a Well [10/09/2001]
A man wishes to determine the water level in a deep well. He drops a
stone into the well and hears it hit the water 3 seconds later...
- Water Main Section Lengths [01/09/2003]
A water main is being laid using pipe that comes in either 18-foot or
20-foot sections. The water main will require 14 fewer sections of 20-
foot pipe than if 18-foot sections are used. Find the total length of
the water main.
- Weighted Average of the Velocities [01/30/2003]
Finding the average velocity for each of the three phases of the
- What are Piecewise Functions? [05/13/2001]
I know piecewise functions are based on expressions between specific
intervals, but I do not know how to describe this function family.
- What are the solutions to x^4+9 = 0 ? [03/31/2003]
How to work with complex numbers when the root is higher than two?
- What Does It Mean to Distribute [08/25/2007]
I understand the basics of the distributive property, but I'm confused
about what the 'over' means, like in the distributive property of
multiplication over addition or over subtraction.
- What is a Partial Fraction? [01/08/1999]
What do partial fractions have to do with polynomials?
- What is the Purpose of a Matrix? [11/04/2003]
What is the purpose of a matrix?
- What's the Second Part? [3/21/1995]
If x = root3 + root2, show that x^4 - 10x^2 +1 = 0. Hence prove that
root3 + root2 is irrational.
- What Time Did Ajay Start? [08/06/2003]
Three persons, Ajay, Bhanu, and Chandu, plan to travel from P and reach
Q at 4.00 PM...
- What We're Doing When We Do Algebra [03/25/2002]
How do you graph x + y = 5 on a graph sheet?
- When FOIL Fails [03/22/2001]
Find the product - (2c-3)3rd power. I can do FOIL on a question to the
second power, but I cannot figure out how to do a question to the third
- When Hands of a Clock Meet [8/29/1996]
To the nearest second, what is the first time after 12:00 that the hour
hand and the minute hand of a clock are together?
- Who Picked the Most? [09/12/1998]
Arrange the names of the people in the order of the number of peaches
that each picked, starting with the person who picked the most.
- Who Will Win the Race? [01/14/1998]
Harry runs 3/4 mile every 6 minutes and Tortisha runs 1/2 mile every 5
minutes. Harry says, "I'll give you a 2 mile head start and still beat
you in the race."
- Why Algebraic Expressions with Parentheses? [03/18/2003]
How would you write these algebraic expressions without parentheses?
-(2x-3y-6) and -(5x-13y-1).
- Why Does Long Division Work? [12/22/2006]
I'm curious why the standard long division algorithm works. How do all
the "bringing down" and other steps lead to the right answer?
- Why Does the Algorithm for Polynomial Long Division Ignore Smaller Terms? [03/24/2011]
A student wonders how polynomial long division works if, at each stage, it disregards terms of smaller degrees. Starting from integer division, Doctor Peterson builds up an example to explain.
- Why Does the Order of Square/Square Root Matter? [8/28/1996]
Why does the square root of x^2 equal absolute value of x, but the square
of the square root of x equals just plain old x?
- Why Do I Have to Learn This? [05/03/2004]
When, in the real world, would I have to find the roots of a quadratic
- Why Do We Have the Order of Operations? [03/05/2006]
Why do we have the order of operations? Why can't we just calculate
from left to right? That seems much easier and more logical to me.
- Why Do We Use Variables in Algebra? [01/31/2004]
Why do we use letters to represent numbers in algebra?
- Why Factor? [08/18/2001]
I've been explained the how's, the where's, the what's, but not the WHY.
What is the practical use of factoring?
- Why Flip the Inequality Sign? [10/26/2001]
Please tell me why you flip the inequality sign when dividing by a
- Why FOIL? [11/21/2002]
How can I explain to the class the importance of learning FOIL?
- Why Some Solutions to Absolute Value Equations are False [10/24/2004]
When solving equations involving absolute value, you sometimes get an
answer that doesn't check in the original equation. What is it about
absolute value that makes this happen?
- Why Use n for Any Number? [09/28/2002]
Why do people use n when any number can fit in or when they want to
- Width and Area of a Field [8/22/1996]
Find a quadratic equation that relates the area of a field to its width.
- Wind Chill Formula [2/1/1996]
Could you tell us the formula used for finding the wind chill?
- Word Problem Regarding Time and Distance [05/30/2002]
A train is going through a tunnel that is 650 feet long. The train
is traveling 100 feet per second. The train enters the tunnel on
one end. Twenty-six seconds later the last car exits the other end
of the tunnel. How long is the train?
- Working with Quadratic Roots [08/15/2002]
Given that a and b are the roots of the quadratic equation x^-5x+3=0,
form, without solving the equation, the quadratic equation whose
roots are a/b and b/a.
- x^3 + y^3 [10/17/2002]
Given: x + y = 1, x^2 + y^2 = 4; x^3 + y^3 = ?
- Young's Formula [08/21/1999]
Calculating a child's dose of medicine based on the adult dose.