Algebra II For Dummies

Algebra Articles

Convert a Circle Equation to the Standard Form

When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle. The equation x² + y² + 6x – 4y – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables. Just follow these steps:

1. Change the order of the terms so that the x's and y's are grouped together and the constant appears on the other side of the equal sign.

   Leave a space after the groupings for the numbers that you need to add:

   x² + 6x _____ + y² – 4y _____ = 3

2. Complete the square for each variable, adding the number that creates perfect square trinomials.

   In the case of the x's, you add 9, and with the y's, you add 4. Don't forget to also add 9 and 4 to the right:

   x² + 6x + 9 + y² – 4y + 4 = 3 + 9 + 4

   When it's simplified, you have x² + 6x + 9 + y² – 4y + 4 = 16

3. Factor each perfect square trinomial.

   The standard form for the equation of this circle is (x + 3)² + (y – 2)² = 16.

Algebra Articles

Solve Rational Inequalities Using the Sign-Line Method

When solving rational inequalities, you use the same steps as for any quadratic inequality. By adding the sign-line method, you can also learn whether the different factors in each interval are positive or negative.

Begin with these steps:

1. Move all the terms to one side of the inequality sign.

2. Factor, if possible.

3. Determine all zeros (roots, or solutions).

   Zeros are the values of the variable that make each factored expression equal to zero.

4. Put the zeros in order on a number line.

5. Create a sign line to show where the expression in the inequality is positive or negative.

   A sign line shows the signs of the different factors in each interval. If the expression is factored, show the signs of the individual factors.

6. Determine the solution, writing it in inequality notation or interval notation.

The signs of the results of multiplication and division use the same rules, so to determine your answer, you can treat the numerator and denominator the same way you treat two different factors in multiplication. Here's an example: solve the rational inequality,

You can use these steps to find the solution for this rational inequality:

1. Every term in

   is to the left of the inequality sign.

2. Neither the numerator nor the denominator factors any further.

3. The two zeros are x = 2 and x = –6.

4. Draw the two numbers on a number line.

5. Create a sign line for the two zeros; you can see here that the numerator is positive when x is greater than 2, and the denominator is positive when x is greater than –6.

6. When determining the solution, keep in mind that the inequality calls for something less than or equal to zero.

   The fraction is a negative number when you choose an x between –6 and 2. You get a negative numerator and a positive denominator, which gives a negative result. Another solution to the original inequality is the number 2. Letting x = 2, you get a numerator equal to 0, which you want because the inequality is less than or equal to zero. You can't let the denominator be zero, though. Having a zero in the denominator isn't allowed because no such number exists. So, the solution of

   In interval notation, you write the solution as (–6, 2].

Algebra Articles

Algebraic Permutations and Combinations

In algebra, you use permutations to count the number of subsets of a larger set. Use permutations when order is necessary. With combinations, you can count the number of subsets when order doesn't matter. The formulas you need are