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jiteshmeghwal9

  • 3 years ago

Hey, guys here is a summary on algebra by me & give ur opinions on that :)

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  1. jiteshmeghwal9
    • 3 years ago
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    \[\Huge{\color{red}{\text{Algebra: Summary}}}\]\[\LARGE{\color{gold}{\star}\color{green}{\text{Quadratic Equation}}}\]The general form of a quadratic equation is \(\color{blue}{ax^2+bx+c=0}\),where \(\color{blue}{a\ne0}\) & \(a, b, c\) are constants.\[\LARGE{\color{green}{\text{solution of quadratic equation}}}\]There are three methods to solve a quadratic equation:- 1. Solving by factoring. 2. Solving by completing the square. 3. Solving by Quadratic formula. \[\LARGE{\color{purple}{\text{Solving by factoring:-}}}\] For solving by factoring, we must use the zero factor principal. Zero factor principal means in a quadratic equation, one side must be left with a zero i.e; \(ax^2+bx+c=0\) so that we can factor left side & can find the roots of the equation. \[\LARGE{\color{purple}{\text{Solving by completing the square:-}}}\] Suppose we have an equation \(ax^2+bx+c=0\), where \(a\ne0\) & a, b, c are constants. Step I- Dividing by the coefficient of \(a\) both sides.\[\dfrac{ax^2+bx+c}{a}=\dfrac{0}{a}\]\[x^2+\dfrac{bx}{a}+\dfrac{c}{a}=0.\]Step II- Transferring \(\dfrac{c}{a}\) to R.H.S.\[x^2+\dfrac{bx}{a}=\dfrac{-c}{a}\]Step III- Completing the square now\[x^2+\left( \dfrac{b}{a} \right)x+\left( \dfrac{b}{2a} \right)^2=\left( \dfrac{b}{2a} \right)^2+\dfrac{-c}{a}\]\[\left( x+\dfrac{b}{2a} \right)^2=\dfrac{b^2}{4a^2}-\dfrac{c}{a}\]\[\left( x+\dfrac{b}{2a} \right)^2=\dfrac{ab^2-4a^2c}{4a^3}\]\[\left( x+\dfrac{b}{2a} \right)^2=\dfrac{\cancel a(b^2-4ac)}{\cancel a(4a^2)}\]\[x+\dfrac{b}{2a}=\sqrt{\dfrac{b^2-4ac}{4a^2}}\]\[x+\dfrac{b}{2a}={\sqrt{b^2-4ac} \over 2a} \]\[x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\]

  2. jiteshmeghwal9
    • 3 years ago
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    \[\Large{\color{purple}{\text{Solving by quadratic formula:-}}}\]This is a very easy method to solve a quadratic equation. Assume u have an equation x^2+2x+4=0. Here, a=1, b=2 & c=4. Therefore by putting this values in the quadratic formula u can easily get \(x\). Note:- Since the quadratic equation's highest degree is 2 therefore its solution also comes in two forms. The main & important part of a quadratic formula is \(\sqrt{b^2-4ac}=D\) which is called discriminant. It helps us to find which type of solution is this. It has the folloing cases:- 1. If D>0, then there are two distinct solutions\[\alpha={-b + \sqrt{b^2-4ac} \over 2a}\]\[\beta=\dfrac{-b - \sqrt{b^2-4ac}}{2a}\]2. If D=0, then the roots are real & equal.\[\alpha={-b+0\over2a}=\dfrac{-b}{2a}\]\[\beta={-b-0\over2a}={-b \over 2a}\]3. If D<0 then the roots are imaginary\[\alpha=\dfrac{-b+\sqrt{-1}}{2a}={-b+\iota \over 2a}\]\[\beta={-b-\sqrt{-1} \over 2a}=\dfrac{b-\iota}{2a}\]

  3. jiteshmeghwal9
    • 3 years ago
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    \[\LARGE{\color{violet}{\text{Linear Equation In two variables}}}\]An equation in the form of \(ax+by+c=0\), \(ax+by+d=0\) where a & b doesn't equal to zero is called a linear equation in two variables. The values of \(x\) & \(y\) satisfying the equation are called the solutions to it. \[\LARGE{\color{violet}{\text{Simultaneous Linear Equations}}}\]A pair of equations in two variables is called simultaneous linear equations in two variables & the values of \(x\) & \(y\) satisfying the equations are called the solutions to it. General form :- \(a_1+b_1=c_1\) \(a_2+b_2=c_2\) (i) \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\) This system has a unique solution. (ii) \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}\) This system has no common solution. (iii) \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\) This system has infinite number of solutions.

  4. jiteshmeghwal9
    • 3 years ago
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    \[\LARGE{\color{blue}{\text{Division Of Two Polynomials:-}}}\] when we divide one polynomial by another polynomial, we got a remainder equals to zero or a number less than the divisor. \(f(x)=g(x).q(x)+r(x)\) where \(g(x)\ne0\) Factor theorem :- If f(x) be a polynomial and 'a' be a real number then (x-a) is a factor of f(x) if f(a)=0. Remainder theorem :- If a polynomial f(x) is divisible by (x-a) then the reminder is f(x) where 'a' is a real number.

  5. jiteshmeghwal9
    • 3 years ago
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    @ParthKohli @experimentX @sauravshakya @hartnn @UnkleRhaukus @satellite73 @myininaya @AccessDenied @radar @Vincent-Lyon.Fr @mathslover have a look :)

  6. jiteshmeghwal9
    • 3 years ago
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    @precal @hba @AravindG have a look :)

  7. AravindG
    • 3 years ago
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    good work @jiteshmeghwal9

  8. TheViper
    • 3 years ago
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    Nice \(\Huge{\color{red}{LaTeX}}\). & the most good thing is that you did so much hard work only to \(teach\). So, you deserved a \(\Huge{\color{green}{Medal}}\). \(\Huge{\color{orange}{\ddot{\smile}}}\)

  9. jiteshmeghwal9
    • 3 years ago
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    Thanx dudes :)

  10. mayankdevnani
    • 3 years ago
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    gr8.....work @jiteshmeghwal9

  11. shubhamsrg
    • 3 years ago
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    there were just 2 silly mistakes from my part.. 1) when you take sqrt of both sides, +- sign comes then only, and not when you finally transfer -b/2a to the other side in solving by completing the square you must be knowing this, i'd say just missed that on typing 2)D is not equal to sqrt(b^2 -4ac) but is b^2 -4ac simply.. otherwise, you've done very good work and i really appreciate your effort ! :)

  12. ParthKohli
    • 3 years ago
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    This is not algebra. It's intermediate algebra to be specific!

  13. UnkleRhaukus
    • 3 years ago
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    Great work @jiteshmeghwal9 ! as shubhamsrg pointed out but you should have the discriminant without the square root (I would use a delta instead of a d) \[\Delta=b^2-4ac\]

  14. jiteshmeghwal9
    • 3 years ago
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    Thanx & i appreciate ur comments & i will improve this mistakes. Thanx to all of u :)

  15. jiteshmeghwal9
    • 3 years ago
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    @ajprincess @Callisto @shadowfiend @yahhave a look :)

  16. jiteshmeghwal9
    • 3 years ago
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    @Yahoo!

  17. jiteshmeghwal9
    • 3 years ago
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    @ghazi @ganeshie8 :)

  18. ghazi
    • 3 years ago
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    this is a good work but i would recommend you to expand it a bit by that i mean to include graph of a quadratic equation and show the solution by curves that will be very helpful for beginners .

  19. jiteshmeghwal9
    • 3 years ago
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    But it will be expanded to calculus & i do not know calculus :(

  20. ghazi
    • 3 years ago
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    hmm no i dont think so, but yea its alright :) i thought you have studied calculus :)

  21. jiteshmeghwal9
    • 3 years ago
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    my standard is class 8 only :)

  22. ghazi
    • 3 years ago
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    then its great :)

  23. jiteshmeghwal9
    • 3 years ago
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    :)

  24. jiteshmeghwal9
    • 3 years ago
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    @Preetha mam have a look :)

  25. whpalmer4
    • 3 years ago
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    You are using "principal" where "principle" should be used. The principle is: If you choose the wrong word, you have to go to the principal's office :-)

  26. ajprincess
    • 3 years ago
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    Great work:) @jiteshmeghwal9

  27. jiteshmeghwal9
    • 3 years ago
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    Ahh..right, lol ;) It is my first big tutorial so little mistakes doesn't matter to me :) Thanx @ajprincess :)

  28. whpalmer4
    • 3 years ago
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    And you've come across another secret: if you want to understand something really well, try to teach it to someone else!

  29. jiteshmeghwal9
    • 3 years ago
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    :)

  30. whpalmer4
    • 3 years ago
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    "Remainder theorem :- If a polynomial f(x) is divisible by (x-a) then the reminder is f(x) where 'a' is a real number." Shouldn't that be the remainder is f(a)?

  31. jiteshmeghwal9
    • 3 years ago
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    :P sorry, i meant that the remainder when f(x) is divided by (x-a) is f(a) where 'a' is a real number

  32. jiteshmeghwal9
    • 3 years ago
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    Ok ! guys now i'm improving my mistakes. \(\color{red}{\text{*Principal=Principle}}\) \(\color{red}{*\sqrt{b^2-4ac}=b^2-4ac}\) \(\color{red}{\text{*Discriminant}=\delta}\) \(\color{red}{\text{*remainder}=f(a), \space \text{not} f(x)}\)

  33. LogicalApple
    • 3 years ago
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    Very well done! I look forward to using your textbook when it gets published!

  34. jiteshmeghwal9
    • 3 years ago
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    Haha, thanx :)

  35. UnkleRhaukus
    • 3 years ago
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    capital Delta \Delta

  36. jiteshmeghwal9
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    Okk !

  37. jiteshmeghwal9
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    @TuringTest have a look :)

  38. jiteshmeghwal9
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    @amistre64 have a look :)

  39. jiteshmeghwal9
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    @eliassaab @Directrix @Diyadiya

  40. Preetha
    • 3 years ago
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    Good work Jitesh!

  41. jiteshmeghwal9
    • 3 years ago
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    Thanx @Preetha mam :)

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