WEEK 10. Systems of equations

1.6x^2 + 1.2x=<-0.2

If you want to be more comfort you should multiply 10 at both sides of equation first.

16x^2 + 12x=<-2

Now you should put everything in one side.

16x^2 + 12x + 2=<0

Here, i choose factoring for solving the inequality:

(4x+1)(4x+2)=<0

You should first find the answer for being 0 and then examine upper and lower of that numbers.

4x=-1

x=-1/4

4x=-2

x=-1/2

IN first case if x will be less than -1/4 the equation will be less than 0 and if x will be more than -1/4 the equation will be more than 0.

In second case if x will be less than -1/2 the equation will be negative and if x will be more than it, the equation will be positive.

So the only zone that the equation is negative is between -1/4 and -1/2.

So the answer is: -1/2=<x=<-1/4

 

week 6.using square roots to solve quadratic equations

In an equation if both sides are square you can solve it by using roots for both sides.

$latex (x+5)^2=64

x+5=+-8

x=8-5=3 or x=-5-8=-13 $

But if one of sides isn’t a complete square you should complete it.

$latex x^2+4x=49

x^2+6x+9=49

(x+3)=+-7

x=-3-7=-10 or x=-3+7=54$

In cases like this you can complete the square by adding the square of half of b.(For example 6 in this case is b)

week5.factoring

12x^2 +4x=4x(3x+1)

For factoring you should first find the same thing between two parts and divide them to it.The exponent of first part is the least exponent in equation.The answer will have two parts,one of them is common part and other one is rest of answer.If you multiply these parts again you should find the same result as first equation if not your answer is incorrect.In below there is an example for bigger exponents:

81x^4+ 27x^3+9x^2=9x^2(9x^2+3x+1)

week 4.Radical equations

\sqrt[4] {256}=(x^{2}-5)^2                                                                                                                                             16=(x^{2}-5)^2                                                                                                                                                               4=x^{2}-5                                                                                                                                                                       9=x^{2}                                                                                                                                                                                    x=3

In this case its easier to find the answer of the left side.The answer of the left side is 16.Now you know that the answer of (x^{2}-5)^2 is 16.In second step you should radical both sides.Now you know that the answer of x^{2}is 9 and after that you can find the x easily.