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

    9 years agoPosted 9 years ago. Direct link to Cian Knight's post “Where does the word "Quad...”

    Where does the word "Quadratic" come from?

    (99 votes)

    • Adithi J

      9 years agoPosted 9 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.)

      (188 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.

      (43 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

      (51 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.

      (23 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

      7 years agoPosted 7 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 quadratic formula Khan Academy? ›

In other words, the quadratic formula is simply just ax^2+bx+c = 0 in terms of x. So the roots of ax^2+bx+c = 0 would just be the quadratic equation, which is: (-b+-√b^2-4ac) / 2a.

When to use a quadratic formula? ›

The quadratic formula is used when solving a quadratic which cannot be factorised.

What is a quadratic equation for dummies? ›

A quadratic equation takes the form ax2 + bx + c = 0 where a, b and c are numbers. The number a cannot be zero. Factorisation and use of the formula are particularly important.

What best describes the quadratic formula? ›

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.

Why do schools teach quadratic formula? ›

The Quadratic Formula is a big deal. If a student can use the Quadratic Formula to solve an equation, we deem them worthy to graduate high school and ready to learn college algebra, the algebra of functions.

What is the point of the quadratic formula? ›

So, what are quadratic equations used for? 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.

How to simplify quadratic formula answers? ›

Step 1: Using inverse operations, move all terms to one side of your equal sign. Step 2: Simplify your equation, and move terms around so that your equation is in the standard form of a quadratic function. Step 3: Now that your equation is in standard form, you can determine the values for a, b, and c.

What are the basic concepts of the quadratic equation? ›

A quadratic equation can be expressed in the general form of ax2+bx+c=0, where a, b and c are numerical coefficients or constants, and the value of x is variable. One fundamental rule of a quadratic equation is that the value of the first constant never can be zero.

What are the 4 ways to solve a quadratic formula? ›

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.

What is the quadratic function formula simplified? ›

Quadratic Functions Formula

The general form of a quadratic function is given as: f(x) = ax2 + bx + c, where a, b, and c are real numbers with a ≠ 0. The roots of the quadratic function f(x) can be calculated using the formula of the quadratic function which is: x = [ -b ± √(b2 - 4ac) ] / 2a.

How to explain the quadratic formula? ›

The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve.

What are 5 methods of solving a quadratic equation? ›

Give 5 ways of solving quadratic equations
  • Extracting the roots.
  • Factoring.
  • Quadratic Formula.
  • Completing the square.
  • Graphing.
Jun 15, 2018

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 a quadratic equation in your own words? ›

Quadratic equations are second-degree algebraic expressions and are of the form ax2 + bx + c = 0. The term "quadratic" comes from the Latin word "quadratus" meaning square, which refers to the fact that the variable x is squared in the equation. In other words, a quadratic equation is an “equation of degree 2.”

What is the meaning of the quadratic formula answer? ›

quadratic formula. noun. : a formula that gives the solutions of the general quadratic equation ax2 + bx + c = 0 and that is usually written in the form x equals 1/2a times (-b ± the square root of b2 - 4ac) Last Updated: 29 Jun 2024 - Updated example sentences.

What is the brief Definition of quadratic equation? ›

Quadratics can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. It is also called quadratic equations. The general form of the quadratic equation is: ax² + bx + c = 0.

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