How to solve rational equations?

Rational Equations: Rational Equations are two polynomial functions written as a ratio. For Example:

Rational equations are just like proportions, which make this quite easy to solve. All you have to do is cross multiply. In the equation above all you have to do is give the fractions a common denominator by multiplying the denominator with the numerator and denominator. After you find its common denominator you cross out the denominator and you are left with the numerator which is x²+6x=8x=24. Now you combine all like terms and you are left with a quadratic equation. Than you solve the quadratic equation and get x=6,-4.

Viola! Simple as that!

Now try this problem on your own:

Sources:

• https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRwefBwxYVBy5zgfpLI96fhYWCofGQjqQFECXTYEK5kP1B5Sjac
• http://www.purplemath.com/modules/rational/solve04.gif

How do we multiply and divide rational expressions?

Multiplying and Dividing rational expressions are really easy. All you need to remember is to multiply straight across then simplify, simple as that! Like this:

Here you factor the expressions out first than you multiplied straight across. 4x+8 over  x²-25 times x-5 over 5x+10. You take factor each expression and you get 4(x+2) over (x+5)(x-5) times x-5 over 5(x+2). Than you cancel out what the expressions have in common like x+2 and x-5. Multiply straight across and viola! the answer is 4 over 5(x+5).

Dividing rational expressions includes a different step. In order to divide rational expressions you must KCF, which stands for Keep Change Flip. You must keep the first expression change the division sign to a multiplication sign than flip the second expression. Like this:

Here you keep the 2x²y over 3yz and change the division sign to a multiplication sign and flip the expression to 16xd²   over 9y²z. Than you cross out what is in common to simplify the expression and multiply straight across, and there is your answer!




Try This:



Sources:

• http://virtualnerd.com/images/Tutorial/Alg1_8_1_4-diagram_thumb-lg.png
• http://www.virtualnerd.com/images/Tutorial/Alg1_08_01_0019-diagram_thumb-lg.png
• http://hotmath.com/help/solutions/genericalg1/7/2/RationalExpressionsandRationalEquations/genericalg1_7_2_RationalExpressionsandRationalEquations_25_605/image062.gif

How do we simplify expressions with rational exponents?

Rational exponents are fractions, for example like this:

The numerator is the number of times the base is to be multiplied. The denominator is the root being taken.
You'll know your expression is simplified when:

1. The exponent isn't negative: You must flip the exponent and put it in the denominator and make it positive.
2. When the index are really simplified: if you can simplify the fraction exponent, do it! Its much easier!

Lets try a Problem:

Here 27 is raised to the 2/3rd power. The denominator is the root, so 27 to the cube root is 3. Than you square 3 and your final answer is 9. Viola! Simple as that.

Now Try this problem on your own:
What is 25/9 to the 1/2 Power?






Sources:

• http://www.solving-math-problems.com/image-files/math_rad_frac_exp.png
• https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQ5GY6vvbHS0glig6_EY2DjtvWd45kOcuOvBBgbKuY9xhD06OR9
• http://www.mathops.com/free/a1rr010.php

How do we rationalize a denominator?

Fractions are quite complicated. The denominator of a fractions is only limited to real integers. Having a radical as a denominator makes the fraction untrue, because it will be an irrational fraction. Therefore, we need to rationalize the denominator. Rationalizing the denominator means to simplify the fraction by getting rid of the radical sign.

How do you rationalize?
Easy! You rationalize the denominator by multiplying the denominator by its conjugate. A conjugate is when you change the sign of the binomial to get rid of the radical. For example:

Here, we need to rationalize the denominator therefore we multiply the√ ̅11 +2 by √ ̅11 -2. Remember whatever you multiply the denominator with you multiply the numerator as well! We multiply 3 by √ ̅11  -2 and viola! Your final answer is 3√ ̅ 11-6/7.


Try This:



Sources:

• http://images.planetmath.org/cache/objects/8052/js/img1.png
• http://tutorial.math.lamar.edu/Classes/Alg/Radicals_files/eq0085M.gif

Factoring by Grouping

Factoring is when the common numbers of a expression or equation is divided out to simplify the expression or equation. Factoring is helpful when solving inequalities, linear inequalities, finding the roots of a quadratic equation and more. However, the easiest way to factor is to factor by grouping.

For example:

In this example, you group 2wx+10w and 7x+35. Than you find the Greatest Common Factor for both of the groupings. You should than get 2w(x+5)+7(x+5). Than you group the like terms which are (x+5) and (2w+7) and viola you are done! The answer is (x+5)(2w+7).

Now, try factoring this: 2x^2-4x+3x-6





Sources:

• http://www.coolmath.com/algebra/04-factoring/images/07-factoring-16.gif

Quadratic Inequalities

Similar to Linear Inequalities, solving them is quite easy. The easiest way to solve them is by graphing the inequality.

For Example In this image above
The quadratic inequality is "Y is less than 2x^2+4x-8" Because it is a less than sign the quadratic is dashed. Than you pick a test point. Above the test point picked is (0,0). You plug in (0,0) in the inequality and if the statement isn't true you shade the outside, if it is true you shade in the parabola.

And that is how you solve Quadratic Inequalities!

Try this:
y > x^2+7x+10




Sources:

• http://algebratesthelper.com/wp-content/uploads/2011/02/graphing-quadratic-inequalities.jpg

Simplifying Imaginary Numbers, yes, Imaginary!

What are Imaginary Numbers?

• Imaginary Numbers are complex numbers, which is in the form of a+bi. A and B are real numbers and i is the square root of -1.

Simplifying Imaginary Numbers are easy. Since the square root of i is -1, simplifying negative square roots are easy. 

For example:

The square root of -9.

To simplify the square root of -9 you must find the square root of -1, which is i.

Than, you find the square root of 9 because the square root of -1 is already found.

The square root of 9 is 3, so finally your answer is 3i.

And that is how you simplify imaginary numbers.