It's really simple!


A quadratic equation consists of three consecutive parts:

-Unknown number squared (Can be x, y and any others)

-Unknown number (Same as squared, but without the square. Example: 2x or 3y)

-Known number.

At the end there should always be =0, that is, we equate the example to 0 for further calculations.



Remember the sequence of numbers, this is important!

To equate a number to zero, you need to move all the numbers after = to the other side, then only 0 will remain.


Example:
X^2 + 2x = 3 > > > X^2 + 2x - 3 = 0

You can transfer any numbers! If you have some calculations after =, like multiplication, which cannot be translated, solve them first.

Example:
X^2 + 2x = 3 * 2 > > > X^2 + 2x = 6 > > > X^2 + 2x - 6 = 0


The most important rule
When you move a number from one side to the other, minus changes to plus, and plus to minus!
In the examples you can see how it works.

If there is no plus or minus in front of the number, then there is a sign + ( + they just don’t usually write )

The number sign always comes before the number itself. For example, if you have the number X - 2, then if you need to write out 2, you write it out with a minus, it turns out -2.

MANDATORY after we have brought the number to 0 (Made it equal to 0), we must calculate the discriminant, it is D .

His formula:

D = b^2 - 4 * a * c

What are these a, b and c, you ask?

It's simple, after we have written a consistent example, we find them.
These letters are known numbers that come before X^2, before X, and also one of them must stand alone.

There is a separate formula for these letters
aX^2 + bX - c = 0


Let's look at an example:

X^2 + 4X - 5 = 0
We calculate numbers using the formula
aX^2 + bX - c = 0


From this it follows:

The number does not appear before X^2, therefore it is multiplied by 1, ( 1 is not written before the number).
You may ask, what does 1 have to do with it? It's simple, when you multiply a number by 1, nothing changes, so we can take and multiply X^2 by 1 so that we have the number a


We found the number at X^2, we are looking for the number in front of the usual X:

We have + 4X in this place, the number in front of X will be equal to 4. We found out that b = 4.


The number that stands apart from the others is -5. We simply rewrite it, it turns out that c = -5.


Now you can calculate the discriminant, just substitute the numbers.
It turns out:

D = b^2 - 4 * a * c
D = 4^2 - 4 * 1 * (-5) = 16 - 4 * (-5) = 16 + 20 = 36

(In a huge number of examples there is such a trick. Since b^2 - 4 * a * c, they often make a or c negative, and when multiplying - on - it turns out to be a plus, in the end we add to b^2.).





You ask, why were we looking for him?

It's simple, it is needed for further calculations X1,2.


Now we find X1 and X2. We find X's. Some of them will stand instead of X^2, and some instead of 4X. We'll find out by the result


For this we need the formula:


X1=(-b+√D) : 2X

X2=(-b-√D) : 2x


Now we substitute the numbers we know.

(I suggest first calculating √D in advance, it’s easier. We have √36 = 6.)


X1=(-4+6) : 2 = 2:2 = 1

X2=(-4-6) : 2 = -10:2 = -5.

This method is a solution to problems without a discriminant and calculations through X



First you need to remember about the formula, or rather about its features
It should be:

1. X^2 must not be negative

2. X^2 should not be multiplied by a number (You can get rid of the number)

3. The discriminant is actually still needed. It doesn't have to be negative (Usually always). If you can't solve the example, check what discriminant you have.




Everything is simpler than with the usual solution.
I will give you an example where you will need to do all three steps.


1. Changing signs:

-2X^2 - 4X + 6 = 0

As we immediately see, it is negative.
We change all the signs to the opposite ones, that is, minus to plus, and plus to minus (I’ll say right away that this function does not work on multiplication and division)


2. Remove the number before X^2:

2X^2 + 4X - 6 = 0

But we see that there is a number in front of X^2, we divide the entire example by 2 so that there is no number.

2X^2 + 4X - 6 = 0 ( :2 )

It turns out:
X^2 + 2X - 3 = 0
We divided all numbers completely by 2.


3. Calculate the discriminant (Not necessary if you are sure that the number fits the theorem)

X^2 + 2X - 3 = 0

We also take numbers:

a = 1
b = 2
c = -3

Calculate the discriminant:

D = 2^2 - 4 * 1 * ( -3 ) = 4 + 12 = 16. Positive, which means we solve the example


( A little hint: If the number a or b is negative, then the calculation will result in b^2 + number, which will definitely give a positive number that will fit for example, because any squared number will always be positive, and minus times minus gives plus. )


4. We calculate using the formula:

Formula:

X1 + X2 = -b
X1 * X2 = c


Let's substitute the numbers:

X1 + X2 = 2
X1 * X2 = -3


From this formula we logically look for X1 and X2, substituting which numbers are suitable:

Let's take the numbers 1 and -3:

It turns out
1 + -3 = 2
1 * -3 = -3

It’s suitable, it means these are our roots.

To solve problems of this type, you need to train in order to quickly calculate it ( I often calculate them in my head, that’s what Vieta’s theorem was made for)

I hope you liked it, if you want to thank me, then like, write comments and (If possible) you can give me community points

Sorry If I made a mistake

Yes, if you are wondering, after finishing school and passing exams, you will never need this again in life. Like 90% of the material in school.







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