x^2 - 5x + 6 = 0
pls explain, need help
x = 2 or x = 3
I literally calculated that in my head. Step up, nigga.
>>8753379
((x + y)^2 - 4xy)/x - y
What about this?
>>8753387
x = 2 or x = 3
I literally calculated that in my head. Step up, nigga.
>>8753390
How did you calculate x when you were given an expression and not an equation?
>>8753374
[math] \displaystyle
\\ ax^2 + bx +c = 0 \; \; \; \; \; \; | \; \cdot 4a \\
4a^2x^2 + 4abx + 4ac = 0 \\
4a^2x^2 + 4abx = -4ac \; \; \; \; \; \; | \; +b^2 \\
4a^2x^2 + 4abx +b^2 = b^2 -4ac \\
(2ax + b)^2 = b^2 -4ac \\
2ax + b = \pm \sqrt{b^2 - 4ac} \\
2ax = -b \pm \sqrt{b^2 - 4ac} \\
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
[/math]
>>8753394
He just found its roots you stupid nigger desu senpai.
>>8753500
Is there an intuitive way to see why you apparently arbitrarily multiply by 4a and add b^2? I understand that it's done to get the factorization to pan out, but how can a non-savant see this?
>>8753520
You can not calculate roots from an expression.
>>8753374
The homework of a 12yo (solve for y)
>>8753554
No. This is how maths work. You just do stuff and hope it all adds up to what you want, no matter how arbitrary it may seem
>>8753554
You don't need to multiply by 4a. See pic related, the superior proof of the quadratic formula.
>>8753554
here's a more intuitive way, using the method of completing the square:
[math] \begin{align*}
ax^2 + bx + c &= 0 \\
x^2 + \frac{b}{a}x &= -\frac{c}{a} \\
x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 &= -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \\
\left(x + \frac{b}{2a}\right)^2 &= \frac{b^2}{4a^2} - \frac{c}{a} \\
\left(x + \frac{b}{2a}\right)^2 &= \frac{b^2 - 4ac}{4a^2} \\
x + \frac{b}{2a} &= \frac{\pm\sqrt{b^2 - 4ac}}{2a} \\
x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{align*} [/math]
>>8753671
>idk how this nigga got x = 2 or x = 3
Really?
Read the first fucking post, faggot
MUH DICK
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>>8753374
X^2-5x+6=0
X(x-5)= -6
X=-6
X-5=-6
X=-1
>>8753554
Adding the extra term or "completing the square" as it is called is actually just a generalization of a special type of trinomial called a perfect square trinomial. When I first learned about perfect square trinomials deriving the quadratic equation was obvious and I can now do it without memorizing it.
Alright.
So consider the trinomial [math]x^2 + 2x + 1[/math]
You should be familiar with it's factors, [math](x+1)(x+1) = (x+1)^2[/math]
Okay, now consider this trinomial...
[math]x^2 + 6x + 9[/math]
What are its factors?
Do you notice anything special about the relation ship between the terms of the trinomial and it's factors? The first trinomial I showed you is a trivial instance of this type of trinomial.
I'll continue...
[math]x^2 + 6x + 9 = (x + 3)(x + 3) = (x+3)^2[/math]
Hmmm.... Is there a more general way to write this? Can we rewrite the numerical coefficients in a more general form?
Well let's see... [math]6 = (2)(3)[/math] ... and [math]3^2 = 9[/math]... So I guess we could rewrite this as
[math]x^2 + (2)(3)x + (3)^2 = (x + 3)(x + 3) = (x+3)^2[/math]
And if replace all the numerical terms that are equal to each other with literal terms (as we do in algebra) we get
[math]x^2 + 2\sqrt c x + c = (x + \sqrt c)(x + \sqrt c) = (x + \sqrt c)^2[/math]
At this point you should definitely see where the extra term comes from... I'll finish explaining anyway.
So as you can see, in general, when the second term of a second degree polynomial (with a = 1) is equal to twice the square root of the third term, c, then you can always factor the expression into [math](x + \sqrt c)^2[/math]
When you derive the quadratic equation, you essentially invent a c, such that c = the square of the coefficint of b so that you can factor the left hand side of the equation into [math](x + \sqrt c)^2[/math], which in the general form of the quadratic equation would look like [math](x + \frac{b}{2a})^2[/math]
>>8754102
>coefficient of b
sorry, coefficient of [math]x[/math] in [math]ax^2 + bx + c[/math]
also all the it's should be its oops.
>>8754007
this post gave me cancer and AIDS, thanks
>>8753374
are you trying to factor or finding the value of x?
also this is literally high-school tier question MODS!!!
>>8753374
x^2-2x-3x+6
when fully factored it becomes (x-3)(x+2)
now i'm guessing you want the zero function which would be x=3 and x=-2
easy as hell t b h you're probably hopeless if you find polynomials hard
>>8754418
(-3)(2)=6
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>>8753554
since x^2+2x+1 is (x+1)^2
then it isn't a far-fetched idea to try to convert
the ax^2+bx into something similar by editing it a bit
>>8754418
fugg made an error, should've been (x-2)(x-3)
which means x=3, 2
Grade 9 maths lmao
>>8753374
>x^2 - 5x + 6 = 0
First you put all the numbers on one side of the equation so
x^x = 0 + 2 - 5 + 6
simplify that to x^x = 256
x triangle x is equal to 2x so 2x = 256 or x = 126. Easy.
Mfw I see people using the ABC formula rather than the reduced, PQ formula.
>>8753705
>((x + y)^2 - 4xy)/x - y
=(x^2+y^2+2xy-4xy)/(x-y)
=(x^2+y^2-2xy)/(x-y)
=((x-y)^2)/(x-y)
=x-y
y=x
>>8753374
I used this method:
[eqn]x^2 +(a+b)x + (a\times b) = 0[/eqn]
and is easy to see that [math]a= -2[/math] and [math]b= -3[/math] so you obtain:
[eqn](x-2)(x-3)=0[/eqn]