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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.
    Mixture problems.
    Quadratic equations.
    Absolute value.
    Completing the square.
    Direct and indirect variation.
    Inequalities and negative numbers.

Solving Rational Equations by Using the LCD [09/27/2007]
How do you solve the equation 3/(n-2) + 6/(n^2-5n+6) = 4/(n-3)?

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 unknowns?

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- 579.59.

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 equation.

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 of Two Squares Equals Twice a Third Square ... Each an Integer? [11/25/2015]
A young adult seeks positive integer solutions to 2a^2 = b^2 + c^2. Doctor Carter obliges by setting up a comparable problem in the Cartesian plane — for points on a circle of radius two — and seeking its rational solutions.

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 higher degree?

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 the water?

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 wobbly?

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.

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