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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.

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.

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 ?

Three Weights [12/07/1997]
A boy selling fruits has only three weights, but with them he can weigh any whole number of pounds from 1 pound to 13 pounds inclusive. What three weights does he have?

Times Greater Than, Times As Much As [05/02/1999]
A number is five times greater than x. Will this number be 6x or 5x?

Top 10 Algebra Mistakes [05/23/1998]
What do you think are the top 10 mistakes that are made in algebra?

A Train and Clock Problem [04/29/2000]
A train leaves the station exactly on the minute. After traveling 10 miles at an average speed of 66 mph, the minute hand is directly over the hour hand. At what time does the train leave the station?

A Train Breaks Down [06/30/2001]
A train breaks down an hour after starting its journey. After repairs, it can only continue at half its original speed and arrives at its destination two hours late...

Trapezoid Vectors [9/9/1995]
In a trapezoid TXYZ, the vector TX = the vector bZY. If the diagonals meet at O, find an expression for the vector TO in terms of the vector TX and the vector TZ.

Traveling Trains and Nonlinear Systems [07/16/1998]
A train leaves town A and travels 540 miles at a uniform speed, followed an hour later by another train.... What were the speeds of the trains?

Triangle inequality [11/14/1994]
Is it possible to prove that the absolute value of (a-b) is greater than or equal to the absolute value of a minus the absolute value of b?

Triangle Perimeter [07/20/1997]
How many triangles have sides whose lengths total 15 units?

Triangular Triples: Means that Are Not So Average [04/28/2012]
A student finds several pairs of triangular numbers that average to a third one, and so wonders how many more such triples exist — and how to generate them. With a few inspired variable substitutions and some modular arithmetic, Doctor Jacques responds, then suggests a few new questions to explore.

A Trick for Solving Equations with Fractions [11/12/1995]
Help solving an equation with fractions.

Trinomial Expansion [12/08/1998]
I need to expand (1+x+x^2)^n to four terms. Is there a formula?

Triples of Numbers [7/23/1996]
Find two triples of positive integers such that their sum equals their product.

Two Airplanes [08/28/1997]
Draw a diagram using coordinate system with origin O - how far apart are the airplanes? Express the answer in radical form.

Two Algebra Word Problems [3/23/1996]
Two people decide to open a music store. A building is available for rent for $6000 per year. Heat, electricity, taxes, etc... are estimated at $5000 per year. Each person wants to make $7 per hour...

Two Cars [08/11/1999]
How long does it take car B to draw even with car A, and how far does car B travel?

Two Equations, Two Unknowns [7/27/1996]
How do I solve for x and y in terms of a and b for the equations x^4 + y^4 = b^4 and x + y = a?

Two Hot-Air Balloons [10/21/1996]
The height h of the first balloon at time t is h(t)=50+34t. The height g of the second balloon is g(t)=400-16t. When will they be at the same height and how high will they be?

Two New Variables [10/25/2002]
I am trying to calculate using the Owens-Wendt equation. The equation will give two equations with two unknowns. I am unable to solve them, because of the x^1/2 and y^1/2 powers.

Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]
Find two numbers such that when you find the sum, product, and quotient of the two numbers they are all equal.

Two Values of X [03/20/2002]
Two numbers differ by 7 and the sum of their squares is 389. Find the two numbers.

Understanding Parallel and Perpendicular [03/23/1998]
Can you explain parallel and perpendicular lines and slopes?

Understanding Rectangle Area and Perimeter [11/08/2002]
True or false: if the perimeter of a rectangle increases, the rectangle's area always also increases.

Unraveling an Inverse Function [11/30/2001]
I got this question wrong: y = (-5x - 2) / (-x + 1).

Unusual Simultaneous Equations [02/06/2003]
A = a*(b+c)/(a+b+c); B = b*(a+c)/(a+b+c); C = c*(a+b)/(a+b+c). A, B, and C are known. Solve for a, b, and c.

Usefulness of General Form of Quadratic Equation [03/24/2005]
When is it useful to know the general form of a quadratic? Why wouldn't I rather know the vertex form or factored form since those tell me something about the graph?

Using Cardan's Formula to find Real Roots [04/13/2000]
How can I show that, although the cubic equation x^3 - 6x = 4 has three real solutions, Cardan's formula can find them by subtracting appropriate cube roots of complex numbers?

Using Euclid's Algorithm with Three Numbers [11/05/2003]
How do I find the GCD of three integers using Euclid's Algorithm? I am not sure where you plug the third integer into the algorithm.

Using Graphs to Solve Problems [12/29/1996]
How do you use graphs to find the answers to questions?

Using Interval Notation to Express Answers [12/03/1996]
Solve for x in the inequality x/(x+3) >= 1/(x-1) and express the answer in interval notation.

Using Relative Primes [12/07/1996]
Given a floor 105 tiles wide and 135 tiles long, how many tiles will a diagonal drawn from one corner to the opposite corner intersect?

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