Quadratic formula explained (article) | Khan Academy (2024)

Gain more insight into the quadratic formula and how it is used in quadratic equations.

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  • Cian Knight

    8 years agoPosted 8 years ago. Direct link to Cian Knight's post “Where does the word "Quad...”

    Where does the word "Quadratic" come from?

    (96 votes)

    • Adithi J

      8 years agoPosted 8 years ago. Direct link to Adithi J's post “Good question! It is deri...”

      Quadratic formula explained (article) | Khan Academy (4)

      Quadratic formula explained (article) | Khan Academy (5)

      Quadratic formula explained (article) | Khan Academy (6)

      Good question! It is derived from the Latin word quadrare, which means "to square", which is what you do in quadratics. Though you may think it means something to do with four, this is not true, because it is simply referring to squaring (a square has four sides.)

      (184 votes)

  • Sam D

    7 years agoPosted 7 years ago. Direct link to Sam D's post “Just curious, is there so...”

    Just curious, is there something like the "Trinomial formula", for third degree polynomials and so on? Or do we figure it out by normal factorization? So what makes second degree polynomials so special over say, 5th, or 3rd degree ones?

    (15 votes)

    • andrewp18

      7 years agoPosted 7 years ago. Direct link to andrewp18's post “Good question!First note...”

      Quadratic formula explained (article) | Khan Academy (10)

      Quadratic formula explained (article) | Khan Academy (11)

      Good question!
      First note, a "trinomial" is not necessarily a third degree polynomial. A trinomial is a polynomial with 3 terms. It can have any degree. A third degree polynomial is called a cubic polynomial. Similar to how a second degree polynomial is called a quadratic polynomial.
      There are general formulas for 3rd degree and 4th degree polynomials as well. These are the cubic and quartic formulas. Both of these formulas are significantly more complicated and difficult to derive than the 2nd degree quadratic formula! Here is a picture of the full quartic formula:
      https://upload.wikimedia.org/wikipedia/commons/9/99/Quartic_Formula.svg
      Be sure to scroll down and to the right to see the full formula! It's huge! In practice, there are other more efficient methods that we can employ to solve cubics and quartics that are simpler than plugging in the coefficients into the general formulae.
      In fact, the highest degree polynomial that we can find a general formula for is 4 (the quartic). The Abel-Ruffini Theorem establishes that no general formula exists for polynomials of degree 5 or higher. So it's not that we haven't yet found a formula for a degree 5 or higher polynomial. It's that we will never find such formulae because they simply don't exist. You can read about the theorem here:
      https://en.wikipedia.org/wiki/Abel–Ruffini_theorem
      So in conclusion, there are only general formulae for 1st, 2nd, 3rd, and 4th degree polynomials. No such general formulas exist for higher degrees.

      (42 votes)

  • Daniel Rendall

    10 years agoPosted 10 years ago. Direct link to Daniel Rendall's post “does x2 = x to the power ...”

    does x2 = x to the power of 2?

    (2 votes)

    • stephen

      10 years agoPosted 10 years ago. Direct link to stephen's post “Yes x with a little 2 to ...”

      Quadratic formula explained (article) | Khan Academy (15)

      Quadratic formula explained (article) | Khan Academy (16)

      Yes x with a little 2 to its top right is x to the power of 2, but for future reference when typing x to the power of 2 on the computer the convention is to use the "^" symbol to say "to the power of"

      so x to the power of 2 would be x^2

      (49 votes)

  • kit wing

    10 years agoPosted 10 years ago. Direct link to kit wing's post “instead of the formula, m...”

    instead of the formula, my textbook wants me to use factorization..how to i do x^2+2x-3=0?
    1. how do i factorize x^2+2-3?
    2. is it possible to use the formula for this? (i tried but cldnt seem to find the answer

    (7 votes)

    • Robert Lee

      10 years agoPosted 10 years ago. Direct link to Robert Lee's post “if you mean find the solu...”

      Quadratic formula explained (article) | Khan Academy (20)

      if you mean find the solution, yes, you would get -3 and 1.
      If you want to factor it would be (x + 3) (x - 1).
      The quadratic formula helps you find the roots not the factored form.

      (22 votes)

  • Anna

    10 years agoPosted 10 years ago. Direct link to Anna's post “Could you extend this qua...”

    Could you extend this quadratic formula to work for other non-linear equations as well? I mean I have heard of so called Octic Equations which are of the form:

    ax^8 + bx^7 + cx^6 + dx^5 + ex^4 + fx^3 + gx^2 + hx + i

    and no I am not using d to mean derivative, or e to mean 2.7... or f, g, and h to mean function of x or i to mean the imaginary unit, just as variables.

    (5 votes)

    • Huron Tu

      8 years agoPosted 8 years ago. Direct link to Huron Tu's post “In 1827, a mathematician ...”

      In 1827, a mathematician by the last name of Abel proved that there is no way to make an analogous equation past the 4th degree. One example (I found all of this on the cubic equation link) is the inverse of the function f(x)=x^5+x. There is simply no way to make an analogous equation for any polynomial of degree y for y>4, not enough operations are defined by the rules of mathematics. Maybe someone who reads this could invent one? : )

      (9 votes)

  • Andy Peter

    a year agoPosted a year ago. Direct link to Andy Peter's post “could we use the quadrati...”

    could we use the quadratic formula when b = 0 or c = 0 ?

    (7 votes)

    • Kim Seidel

      a year agoPosted a year ago. Direct link to Kim Seidel's post “Yes, you can use the quad...”

      Quadratic formula explained (article) | Khan Academy (27)

      Yes, you can use the quadratic formula for all quadratic equations.

      (11 votes)

  • Karyn Williams

    7 years agoPosted 7 years ago. Direct link to Karyn Williams's post “I do not enjoy math and I...”

    I do not enjoy math and I need some help.

    (1 vote)

    • nkfonseka

      6 years agoPosted 6 years ago. Direct link to nkfonseka's post “Start from the beginning ...”

      Quadratic formula explained (article) | Khan Academy (31)

      Start from the beginning of Khan Academy. Work through it Lesson by Lesson. Make sure not to skip any lessons or videos. This might help.

      (23 votes)

  • Nafia Farzana

    7 years agoPosted 7 years ago. Direct link to Nafia Farzana's post “How do i know when the cu...”

    How do i know when the curve goes like a u or a upside down u ?

    (6 votes)

    • Estelle Pretorius

      7 years agoPosted 7 years ago. Direct link to Estelle Pretorius's post “If the coefficient of x^2...”

      Quadratic formula explained (article) | Khan Academy (35)

      If the coefficient of x^2 is negative, the curve will look like an upside down u (i.e. the curve will have an absolute maximum). If the coefficient of x^2 is positive, the curve will look like a u (i.e. the curve will have an absolute minimum).

      Hope this helps.

      (13 votes)

  • Patrick

    8 years agoPosted 8 years ago. Direct link to Patrick's post “For the quadratic formula...”

    For the quadratic formula, I have a quick question. For the b^2 part inside the square root, why can't it be transferred to the outside as b?

    (4 votes)

    • MBlackwll

      8 years agoPosted 8 years ago. Direct link to MBlackwll's post “Hopefully this proof help...”

      Hopefully this proof helps you understand why:
      https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-using-the-quadratic-formula/v/proof-of-quadratic-formula

      (6 votes)

  • almadugomez

    9 years agoPosted 9 years ago. Direct link to almadugomez's post “how is the quadratic form...”

    how is the quadratic formula derived

    (3 votes)

    • Just Keith

      9 years agoPosted 9 years ago. Direct link to Just Keith's post “There are several ways to...”

      There are several ways to derive the quadratic formula, but the simplest is by using completing the square.
      https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-using-the-quadratic-formula/v/proof-of-quadratic-formula

      (8 votes)

Quadratic formula explained (article) | Khan Academy (2024)
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