How do we solve quadratic inequalities?

How do we solve quadratic inequalities? 

Okay so for this question there are two ways to solve.

There's algebraically and graphically. Let's just do algebraically because its the hardest, well at least to me it is. 

When you get an inequality, the other side to the inequality is going to be 0. 

Now what is most important it when you factor an inequality. Once you do that, that is what determines the answer and the symbol of each answer(s) inequality. 

For example:

x² - x - 6 < 0

So now you's factor that and get:

(x - 3) (x + 2) < 0

Now here's the trick to remember. When your symbols are different, the product on the other side has to be negative. 

But back to the point, since the symbols are different this means that the answers will has different inequality symbols. 

Let's see what our answers would be. 

x > -2 and x < 3

You could also test this with 0. 

If you were to graph this it would be an "and" solution.

This is going to equal -2 < x < 3

So the area in between the black lines
would be where the inequality
is true. 

Sources:

1. John Schnatterly johnschnatterly.blogspot.com
2. http://www.regentsprep.org/Regents/math/algtrig/ATE6/Quadinequal.htm
3. http://www.mathsisfun.com/algebra/inequality-quadratic-solving.html

How do we solve quadratic equations by completing the square?

How do we solve quadratic equations by completing the square? 

Well let's discuss this shall we. 

When you see a quadratic equation you should or already think of an equation like this: 

But instead having the a, b, and c
substituted as numbers.

Now using the "completing the square" trick is as easy as it sounds. Actually exactly as it sounds. 

Let's say our equation is x² - 4x + 5

Now to solve this equation this way, you would start off by subtracting 5 from both sides. 

x² - 4x + 5

           -  5

x² - 4x = -5

So now after you do that, you take the value for b (if you forgot which number that is look at the picture above) and divide by 2. 

Once you've done that take that answer and square it. That gives you the new square that is know the value for c. 

x² - 4x = -5

(-4 divided by 2 is -2)

(and -2 squared is 4)

(so you would add 4 to each side)

x² - 4x = -5

     +4    +4

x² - 4x + 4 = -1

And that's simply how you get the new equation to solve this equation. 

But this is the most important step to solving a quadratic equation by completing the square. 

It's that simple. Now if you want to keep on and solve the equation, be my guest ! :)

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Okay i'll do one. But i'll keep doing the same one !

Now you would factor the equation, right. 

x² - 4x + 4 = -1

(both -2 and -2 add to -4 and multiply to 4)

( x - 2 )² = -1

[the factor is squared because it would be multiplied by the same factor]

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SO now you would square root both sides to get rid of the squared factor. 

√( x - 2 )² = √-1

(now that leaves x-2 by itself)

x-2 = √-1

And i would say that this is where it gets complicated. Here you subtract 2 to each side. 

x-2 = √-1

-2    -2

(now you'd put the number subtracted infront an use the ± symbol)

x = -2 ± √-1

& that'd be your answer !!! Hope this helped.

Sources:

1. John Schnatterly johnschnatterly.blogspot.com
2. http://www.google.com/imgres?um=1&hl=en&sa=X&tbo=d&biw=1163&bih=540&tbm=isch&tbnid=l53rkXgBP6c6qM:&imgrefurl=http://www.dace.co.uk/al_khwarizmi.htm&docid=V5Ur7Da2awEgzM&imgurl=http://www.dace.co.uk/math/quadratic_eq.gif&w=298&h=131&ei=IpGoUNjPFOPA0AGUyYDQAg&zoom=1&iact=rc&dur=2&sig=102896554668256793387&page=1&tbnh=104&tbnw=230&start=0&ndsp=17&ved=1t:429,r:2,s:0,i:77&tx=140&ty=54