Quadratic equation

[Yuseph]

Yo guys,

Im finished with the quadratic equations. But i have a couple of questions.
- i suppose you guys dont use the factorization method. it looks like its meant for total beginners. So what dyou do instead. Completing the square or the quadratic formula or graphs ?
- there re different levels of equations. And correct me if im wrong. Heres what i understood so far. Simple equation is when you have one unknown and this unknow can take only one value. You only need one equation to solve it. The only skill required is knowing cross multiplication. Simultaneous equation is when you have 2 unknown. You need two proportionnal equations to solve it otherwise its not valid. Both unknow can only take one value each. Then you have quadratic equation. Whenever you see a squared unknown thats the method you should apply. And the power raised can only be power 2 no more. Now im too much of a beginner to figure out why when theres no y involved x can takes two values. Its just too new for me to be able to explain it to myself.
- looking at the picture can you tell me why the denominator is a instead of 2a, is it a typo ? Because if thats true im totally lost. Also looking at the right hand side can you tell me how the author got to b^2 - 4ac divided by 4a^2. Ive never seen something like that since i started. How can you just move c/a like that and put it on top ?

Thanks. after that ill be working on logarithm i think this is the chapter where i can start fooling people and make them believe im smarter than they imagined even if nothing changed hehe

 

[Dr_Ochrome]

First of all, l didn't understand what you called simultaneous equations.
According to quadratic equation, there some methods using graphs(you can take a look on some YouTube videos about it) but it's better to use quadratic formulas given in your lesson.
In the line you marked in yellow, just develop each side of each equality and you'll find that they are the same.
To get used to those formulas, apply it in some examples and it'll become easier.

 

[JeffM]

My goodness.

First, within the last ten days you were told that cross multiplication applied only to special cases. It is not the only technique required to solve every linear equation. Nor is a system of equations solvable only if it involves two proportions. Where are you getting these ideas?

Second, I use the factorization method when the factoring is obvious to me. Otherwise I use the quadratic formula. I see no practical reason to waste time looking for a factorization or completing the square.

It seems to me that you are looking for some generalizations about what you are learning.

There are many ways to categorize equations, but probably the most important one is the binary categorization between those that are analytically solvable and those that are not. In algebra we study only those that are analyticaly solvable.

If we have n unknowns, we must have at least n pieces of information to find numerical solutions. What is most advantageous is to have n independent equations. I do not see much value in categorizing equations by the number of unknowns. The general method for solving a system of n equations is by substitution, which involves reducing the system by one unknown at a time until you arrive at a single equation in one unknown.

Linear equations and quadratic equations are types of polynomial equation. Respectively, they are of degree 1 and 2. A Polynomial of degree n has at most n distinct solutions. If the degree is odd, a polynomial with real coeffients has at least one real solution; if the degree is even, it may not have any real solution. If the degree exceeds 4, it may not be solvable by the methods taught in algebra.

 

[Yuseph]

Ok
By cross multiplying i mean a/d = c equals d/a = c equals c = ad not a/b = c/d equals ac = bd. i dont know if theres a specific name to call those 3 interchangeables unknowns.
In simultaneous equations. At least in the physics examples ive been given. For the problem to be solved there had to be a law of proportion between the two. Like one equation define one scenario the 2nd equation define another scenario but both based on the same law. Anyway I can only guess by what ive read so far.
Ok ill classify by degree then. I hope i wont need anything higher than 2nd degree in electronics.
Any idea about the fraction in the picture ?

 

[Yuseph]

Ok the answer is obvious as usual.

-4ac/4a^2 + b^2/4a^2
Sorry about this question

 

[R.M.]

Let's walk through that derivation with a little more detail.
ax²+ bx + c = 0
x²+ [MATH]\frac{b}{a}[/MATH]x = -[MATH]\frac{c}{a}[/MATH] --> subtract c from both sides and divide each term by a
[MATH]x^{2} + \frac{b}{a}x + \frac{b^{2}}{(2a)^{2}}[/MATH] = -[MATH]\frac{c}{a}[/MATH] + [MATH]\frac{b^{2}}{(2a)^{2}}[/MATH] --> add this term to both sides to 'complete the square.
(x + [MATH]\frac{b}{2a}[/MATH])² = -[MATH]\frac{c}{a}[/MATH] + [MATH]\frac{b^{2}}{4a^{2}}[/MATH] --> factor left side
(x + [MATH]\frac{b}{2a}[/MATH])² = -[MATH]\frac{4ac}{4a^2}[/MATH] + [MATH]\frac{b^{2}}{4a^{2}}[/MATH] --> create common denom. by multiplying by 4a to fraction on left.
(x + [MATH]\frac{b}{2a}[/MATH])² = [MATH]\frac{b^2-4ac}{4a^2}[/MATH] --> combine fractions and rearrange terms.
x + [MATH]\frac{b}{2a}[/MATH] = [MATH]\pm\sqrt{\frac{b^2-4ac}{4a^2}}[/MATH] --> square root of both sides.
x + [MATH]\frac{b}{2a}[/MATH]= [MATH]\pm{\frac{\sqrt{b^2-4ac}}{2a}}[/MATH] --> simplify denominator.
x = [MATH]{\frac{-b\pm\sqrt{b^2-4ac}}{2a}}[/MATH] --> subtract b/2a from both sides. Done.

 

[JeffM]

Ok
By cross multiplying i mean a/d = c equals d/a = c equals c = ad not a/b = c/d equals ac = bd. i dont know if theres a specific name to call those 3 interchangeables unknowns.
In simultaneous equations. At least in the physics examples ive been given. For the problem to be solved there had to be a law of proportion between the two. Like one equation define one scenario the 2nd equation define another scenario but both based on the same law. Anyway I can only guess by what ive read so far.
Ok ill classify by degree then. I hope i wont need anything higher than 2nd degree in electronics.
Any idea about the fraction in the picture ?

I'm not qualified to answer questions about physics so I do not know whether all equations in physics are that simple.

I do not want to mislead you. Polynomials are just one kind of equation. They have a number of nice properties, and they arise in numerous practical situations. But they are not the only kind of equation with practical uses, and other kinds may not have degree.