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?

    (97 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.)

      (185 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)

      Quadratic formula explained (article) | Khan Academy (17)

      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

      (50 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 (21)

      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 ...”

      Quadratic formula explained (article) | Khan Academy (25)

      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? : )

      (10 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 ?

    (8 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 (29)

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

      (12 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 ?

    (7 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 (33)

      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.

      (14 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 (37)

      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)

  • 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)

FAQs

What is the simple explanation of the quadratic formula? ›

In math, we define a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. The standard form of a quadratic is y = ax^2 + bx + c, where a, b, and c are numbers and a cannot be 0. Examples of quadratic equations include all of these: y = x^2 + 3x + 1.

When to use a quadratic formula? ›

The quadratic formula can be used to solve any quadratic equation but is best saved for when an equation cannot be factorised.

How to solve quadratic functions? ›

Set the equation equal to zero. If the quadratic side is factorable, factor, then set each factor equal to zero. If the quadratic equation involves a SQUARE and a CONSTANT (no first degree term), position the square on one side and the constant on the other side. Then take the square root of both sides.

What is an example of a quadratic equation? ›

Examples of the standard form of a quadratic equation (ax² + bx + c = 0) include: 6x² + 11x - 35 = 0. 2x² - 4x - 2 = 0. -4x² - 7x +12 = 0.

What is a quadratic equation for dummies? ›

A quadratic equation is any second degree polynomial equation — that's when the highest power of x, or whatever other variable is used, is 2. You can solve quadratic equations by factoring. Bring all terms to one side of the equation, leaving a zero on the other side.

What best describes the quadratic formula? ›

Definition of quadratic equation

A quadratic equation is a second order equation written as ax2 + bx + c = 0 where a, b, and c are coefficients of real numbers and a ≠ 0.

What do you use the quadratic formula for in real life? ›

Quadratic equations are used in many real-life situations such as calculating the areas of an enclosed space, the speed of an object, the profit and loss of a product, or curving a piece of equipment for designing.

Why is the quadratic formula so useful? ›

In real life, the quadratic formula helps us in determining the area of space, the speed of a moving object, the value of profit gained on a product, and more. Even the path of a space rocket is described in terms of a quadratic equation.

What is the point of the quadratic formula? ›

The purpose of the quadratic formula is to give us the roots of the parabola and in this form, we can see that the two roots are equidistant from the axis of symmetry h. One root is to the left, and the other root is the same amount to the right.

What are the 4 methods of solving quadratic equations? ›

Answer: There are various methods by which you can solve a quadratic equation such as: factorization, completing the square, quadratic formula, and graphing. These are the four general methods by which we can solve a quadratic equation.

How is the quadratic formula derived? ›

The quadratic formula, x=- b±sqrt(b^2-4ac)/2a, can be derived by completing the square on the general standard form a quadratic equation. Recall that completing the square is a method for solving quadratic equations.

How to teach quadratic equations in a fun way? ›

Have students create a video of themselves solving a quadratic equation using one method. You can allow students to choose or you can tell them to use the quadratic equation, factoring, or completing the square. Students must ask like they are the tutor and explain each step.

How can you tell if something is a quadratic function? ›

f(x) = ax2 + bx + c, where a, b, and c are real numbers. The very definition of a quadratic function explains how to identify if a given function is quadratic. That is, if the highest exponent of the function is 2 and it can be put in the form f(x) = ax2 + bx + c, then it's a quadratic function.

What is a quadratic function easy explanation? ›

A quadratic function is defined as a polynomial where the highest degree of any variable is 2. In other words, a term in the equation will have an exponent to the power of 2. An equation such a f ( x ) = x 2 + 4 x − 1 would be an example of a quadratic function because it has x to the second power as its highest term.

What is the quadratic equation simplified formula? ›

The quadratic equation formula to solve the equation ax2 + bx + c = 0 is x = [-b ± √(b2 - 4ac)]/2a. Here we obtain the two values of x, by applying the plus and minus symbols in this formula. Hence the two possible values of x are [-b + √(b2 - 4ac)]/2a, and [-b - √(b2 - 4ac)]/2a.

What is the quadratic formula for kids? ›

quadratic equations contain terms with powers no higher than two. often in the form ax2 + bx + c = 0 where x is the variable; a, b and c are constants and a ≠ 0.

What is a quadratic equation easy? ›

It will have a squared unknown term and this will be the highest power of the unknown - this is what qualifies it as quadratic. Examples of quadratic equations are: x2+10x=-16.

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