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Understanding the quadratic formula

Gain more insight into the quadratic formula and how it is used in quadratic equations.
The quadratic formula helps you solve quadratic equations, and is probably one of the top five formulas in math.  We’re not big fans of you memorizing formulas, but this one is useful (and we think you should learn how to derive it as well as use it, but that’s for the second video!).
If you have a general quadratic equation like this:
ax2+bx+c=0
Then the formula will help you find the roots of a quadratic equation, i.e. the values of x where this equation is solved.

The quadratic formula

x=b±b24ac2a
It may look a little scary, but you’ll get used to it quickly!
Practice using the formula now.

Worked example

First we need to identify the values for a, b, and c (the coefficients). First step, make sure the equation is in the format from above, ax2+bx+c=0:
x2+4x21=0
  • a is the coefficient in front of x2, so here a=1 (note that a can’t equal 0 -- the x2 is what makes it a quadratic).
  • b is the coefficient in front of the x, so here b=4.
  • c is the constant, or the term without any x next to it, so here c=21.
Then we plug a, b, and c into the formula:
x=4±1641(21)2
solving this looks like:
x=4±1002=4±102=2±5
Therefore x=3 or x=7.

What does the solution tell us?

The two solutions are the x-intercepts of the equation, i.e. where the curve crosses the x-axis. The equation x2+3x4=0 looks like:
Graphing quadratic equations
where the solutions to the quadratic formula, and the intercepts are x=4 and x=1.
Now you can also solve a quadratic equation through factoring, completing the square, or graphing, so why do we need the formula?
Because sometimes quadratic equations are a lot harder to solve than that first example.

Second worked example

Let’s try this for an equation that is hard to factor:
3x2+6x=10
Let’s first get it into the form where all terms are on the left-hand side:
(3)ax2+(6)bx+(10)c=0
The formula gives us:
x=6±62431023=6±361206=6±846
We know you can’t take the square root of a negative number without using imaginary numbers, so that tells us there’s no real solutions to this equation.  This means that at no point will y=0, the function won’t intercept the x-axis.  We can also see this when graphed on a calculator:
Quadratic equation shown on a calculator
Now you’ve got the basics of the quadratic formula!
There are many more worked examples in the videos to follow.

Tips when using the quadratic formula

  • Be careful that the equation is arranged in the right form: ax2+bx+c=0 or it won’t work!
  • Make sure you take the square root of the whole (b24ac), and that 2a is the denominator of everything above it
  • Watch your negatives: b2 can’t be negative, so if b starts as negative, make sure it changes to a positive since the square of a negative or a positive is a positive
  • Keep the +/ and always be on the look out for TWO solutions
  • If you use a calculator, the answer might be rounded to a certain number of decimal places. If asked for the exact answer (as usually happens) and the square roots can’t be easily simplified, keep the square roots in the answer, e.g. 2102 and 2+102

Next step:

Khan Academy video wrapper
Using the quadratic formulaSee video transcript
  • Prove the quadratic formula:
Khan Academy video wrapper
Proof of the quadratic formulaSee video transcript

Want to join the conversation?

  • starky tree style avatar for user Cian Knight
    Where does the word "Quadratic" come from?
    (90 votes)
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    • aqualine ultimate style avatar for user Adithi J
      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.)
      (169 votes)
  • hopper jumping style avatar for user Sam D
    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?
    (12 votes)
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    • mr pink red style avatar for user andrewp18
      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.
      (33 votes)
  • male robot hal style avatar for user Daniel Rendall
    does x2 = x to the power of 2?
    (4 votes)
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    • leaf green style avatar for user stephen
      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
      (46 votes)
  • starky ultimate style avatar for user kit wing
    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)
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  • female robot grace style avatar for user Anna
    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.
    (6 votes)
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    • piceratops ultimate style avatar for user Huron Tu
      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)
  • male robot hal style avatar for user Andy Peter
    could we use the quadratic formula when b = 0 or c = 0 ?
    (8 votes)
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  • starky sapling style avatar for user Elaine
    Honestly, this is pretty easy and quite fun.
    (12 votes)
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  • female robot ada style avatar for user Nafia Farzana
    How do i know when the curve goes like a u or a upside down u ?
    (5 votes)
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  • blobby green style avatar for user Karyn Williams
    I do not enjoy math and I need some help.
    (0 votes)
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  • blobby green style avatar for user chakraborty.sanchari
    I am glad to see that this comment section is not from 8 years ago lol
    (10 votes)
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