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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.
- Solving Rational, Radical, and Polynomial Inequalities [12/28/2003]
I'm struggling with polynomial inequalities, such as (x - 2)(x + 3) >
0 or (x - 2)/(x + 3) < 2. I'm also not sure how to do inequalities
that have square roots in them. Can you help?
- Solving Sqrt(x + Sqrt(x + Sqrt(x + ... = 10 [02/26/2001]
How can I solve the equation sqrt(x+sqrt(x+sqrt(x+...=10 for x if x is
known to be a real number?
- Solving Systems of Equations [10/30/1996]
Solve for a, b, and c: 2a+6b+c = 5, a+b-c = -1, a+8b+2c = 0.
- Solving Systems of Linear Equations [02/09/1999]
Can you help me solve simultaneous equations?
- Solving Systems of Three Equations [05/07/1998]
Can you help me solve the following system of three equations with three
- Solving x^3 + y^3 + z^3 [02/22/2003]
Given: x + y + z = 0; x * y * z = 2; x,y,z are not equal to each
other; x,y,z all real numbers.
- A Sphere's Surface Area and Volume [12/17/1998]
What is the relation between a sphere's surface area and its volume? How
does their ratio change?
- Spiral Problem [02/03/2003]
Given an x,y plane, with the number zero at the center, so it has
coordinates (0,0), if given the coordinates, is there a formula to
find the number that will be at those coordinates?
- Sqrt (x-2) = Sqrt (x+2) ? [01/10/2003]
Is the following equation solvable for x? sqrt (x-2) = sqrt (x+2)
- Squares of Positive Integers [8/13/1996]
The positive integers a and b: the numbers 15a+16b and 16a-15b are both
squares of positive integers...
- State the Value of K [11/26/1997]
State the value of k so the given point lies on the graph of the
equation 2x + ky = 21.
- Straight Line on the Graph [09/10/1998]
Let f(x) = ae^(kx) where a>0 and k are constants. Show that the graph of
y = ln(f(x)) is a straight line.
- Strategy: Finding a Formula [11/14/1997]
Jill needs three pieces of lumber a total of 27 ft. long for a dog pen.
The second piece has to be 4 ft. longer than the first and the third 4
ft. longer than the second...
- Substituting Parameters for Numbers [11/29/2001]
I don't know what to do with both sides of the equation 17.20X/100 = X-
- Substitution to Eliminate Linear Terms [2/9/1996]
Are there any substitutions for eliminating the linear term in both
quadratic and quartic equations?
- Subtracting Different Types of Units [05/26/2001]
When we subtract equations when solving a word problem, it looks as if we
are subtracting the number of items from the cost of the items. How can
we subtract different units?
- Subtracting One Equation from Another [05/30/2001]
Why can you subtract one equation from another and get something
meaningful that you can use to solve the equations?
- Summing the Integers from 1 to n [08/31/2005]
How do you add up the integers from 1 to n by using a formula instead
of adding them all up the long way?
- Sum of a Number and Its Reciprocal [01/28/2002]
The sum of a number and its reciprocal is 5 1/5. What is the number?
- Sum of a Pair — and of the Pair's GCF and LCM [12/09/2012]
A student struggles to identify a pair of positive integers, given their sum as well as the
sum of their greatest common factor and least common multiple. Doctor Greenie
applies some algebra and factorization to turn the problem into a Diophantine
- The Sum of the First N Squares [02/20/1998]
Find a formula for the sum of the first N squares (i.e. 1^2 + 2^2 + 3^
2+.....+N^2) by mathematical induction.
- The Sum of Two Numbers is 20... [11/03/1997]
Twice one number is 4 more than four times the other. Find the numbers.
- Sum or Difference of Two Cubes [10/16/1997]
Factor and check: c^3 + d^3; b^3 - c^3; x^3 + 8; y^3 - 125.
- Sums and Products of Roots of Polynomials [07/24/2002]
In a quadratic equation, the sum of the roots is -b/a and the product
of the roots is c/a. Do similar relations exist for equations of
- Swimming Laps [04/15/2002]
John decides to swim a certain number of laps of the pool in five
days. On the first day he covers one fifth of the total. The next day
he swims one third of the remaining laps...
- Swimming Upstream and Down [09/19/1997]
A man swims 200 meters upstream in 3 minutes (180 seconds) and downstream
150 meters in .7 minutes. How fast is the man swimming and how fast is
- Symbol for Irrational Numbers? [09/23/2002]
What is the standard symbol used to represent the irrational numbers?
Is it Q-bar?
- Synthetic Division [08/05/1997]
Please explain synthetic division.
- Synthetic Division in Factoring [04/18/2002]
Factor completely: 2x^3 + 17x^2 + 58x + 25.
- System of Equations and the Substitution Method [06/07/1998]
Can you explain the "substitution" method of solving systems of
equations, and provide an example?
- Tangent to Parabola [10/21/1996]
What is the slope of the lines tangent to the parabola y = x^2 that pass
through the point (2,1)?
- Teaching Inequalities [04/01/2009]
We're looking for the best way to help students understand the idea of
greater than or equal to, including some real-life examples of how it
might be used. Do you have any thoughts?
- Terminology: Quadratic Equation vs. Quadratic Function [04/07/2004]
As I read various algebra books for high school kids, I find what
appears to be an inconsistent use of the words 'quadratic equation',
and I wanted to be sure I use it correctly myself. Is it correct to
call y = ax^2 + bx + c a 'quadratic equation', or a 'quadratic function'?
- Theorem About Sum and Product of Quadratic Roots [03/07/1998]
Explain why 1/2 and 3/4 are NOT the roots of 0 = 4x^2 + 5x + 8.
- Thinking about the Mediant of Two Fractions [12/10/2009]
One of my students noticed that when given 1/a and 1/b, the fraction
2/(a+b) always seems to be in between 1/a and 1/b. Is this a new
fraction rule, or has this been discovered and proved before?
- Thoughtful Trial and Error [05/21/2002]
Arrange the numbers 1 through 10 in a triangle so that all the rows of
three numbers have the same sum, and all the rows of four numbers have
the same sum.
- A Three-Legged Stool [06/26/2001]
Why is a three-legged stool steady, while a four-legged stool can be
- Three Points, Two Unknown x-coordinates, All in a Row? [03/04/2013]
Given three points, two with unknown abscissas, a student has doubts about their
collinearity. After offering some leading hints, Doctor Mike invokes the point-slope
form to rigorously confirm the student's suspicions.
- Three Sample SAT Questions [11/12/1996]
How do you find the length of diagonal of a square? If (x+2)^2=25 and x>0
what is the value of x^2? Find the value of the denominator that ensures
that a set of fractions is in reduced form.
- Three Ways to Solve Quadratic Equations [05/01/2003]
How do I solve x^2 + 4x + 2 = 0 ?