Where does the word "Quadratic" come from?

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.)

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?

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.

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

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.

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.

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? : )

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

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

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

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.

does x2 = x to the power of 2?

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

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?

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

I do not enjoy math and I need some help.

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.

Main content

Understanding the quadratic formula

CCSS.Math:

HSA.REI.B.4

HSA.REI.B.4b

Google Classroom

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:

a, x, squared, plus, b, x, plus, c, equals, 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, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction

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, a, x, squared, plus, b, x, plus, c, equals, 0:

x, squared, plus, 4, x, minus, 21, equals, 0

a is the coefficient in front of x, squared, so here a, equals, 1 (note that a can’t equal 0 -- the x, squared is what makes it a quadratic).
b is the coefficient in front of the x, so here b, equals, 4.
c is the constant, or the term without any x next to it, so here c, equals, minus, 21.

Then we plug a, b, and c into the formula:

x, equals, start fraction, minus, 4, plus minus, square root of, 16, minus, 4, dot, 1, dot, left parenthesis, minus, 21, right parenthesis, end square root, divided by, 2, end fraction

solving this looks like:

\begin{aligned} x&=\dfrac{-4\pm\sqrt{100}}{2} \\\\ &=\dfrac{-4\pm 10}{2} \\\\ &=-2\pm 5 \end{aligned}

Therefore x, equals, 3 or x, equals, minus, 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 x, squared, plus, 3, x, minus, 4, equals, 0 looks like:

where the solutions to the quadratic formula, and the intercepts are x, equals, minus, 4 and x, equals, 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:

3, x, squared, plus, 6, x, equals, minus, 10

Let’s first get it into the form where all terms are on the left-hand side:

start underbrace, left parenthesis, 3, right parenthesis, end underbrace, start subscript, a, end subscript, x, squared, plus, start underbrace, left parenthesis, 6, right parenthesis, end underbrace, start subscript, b, end subscript, x, plus, start underbrace, left parenthesis, 10, right parenthesis, end underbrace, start subscript, c, end subscript, equals, 0

The formula gives us:

\begin{aligned} x&=\dfrac{-6\pm\sqrt{6^2-4\cdot 3\cdot 10}}{2\cdot 3} \\\\ &=\dfrac{-6\pm\sqrt{36-120}}{6} \\\\ &=\dfrac{-6\pm\sqrt{-84}}{6} \end{aligned}

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, equals, 0, the function won’t intercept the x-axis. We can also see this when graphed 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: a, x, squared, plus, b, x, plus, c, equals, 0 or it won’t work!
Make sure you take the square root of the whole left parenthesis, b, squared, minus, 4, a, c, right parenthesis, and that 2, a is the denominator of everything above it
Watch your negatives: b, squared 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 plus, slash, minus 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. start fraction, 2, minus, square root of, 10, end square root, divided by, 2, end fraction and start fraction, 2, plus, square root of, 10, end square root, divided by, 2, end fraction

Next step:

Practice using the quadratic formula.
Watch Sal do an example:

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

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Cian Knight
Posted 7 years ago. Direct link to Cian Knight's post “Where does the word "Quad...”
Where does the word "Quadratic" come from?
Button navigates to signup pageComment on Cian Knight's post “Where does the word "Quad...”
(79 votes)
Answer
- Adithi J
  Posted 7 years ago. Direct link to Adithi J's post “Good question! It is deri...”
  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.)
  Comment on Adithi J's post “Good question! It is deri...”
  (156 votes)
Daniel Rendall
Posted 9 years ago. Direct link to Daniel Rendall's post “does x2 = x to the power ...”
does x2 = x to the power of 2?
Button navigates to signup pageComment on Daniel Rendall's post “does x2 = x to the power ...”
(4 votes)
Answer
- stephen
  Posted 9 years ago. Direct link to stephen's post “Yes x with a little 2 to ...”
  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
  Comment on stephen's post “Yes x with a little 2 to ...”
  (41 votes)
Sam D
Posted 6 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?
Button navigates to signup pageButton navigates to signup page
(12 votes)
Answer
- andrewp18
  Posted 6 years ago. Direct link to andrewp18's post “Good question! First note...”
  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.
  Comment on andrewp18's post “Good question! First note...”
  (23 votes)
kit wing
Posted 9 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
Button navigates to signup pageComment on kit wing's post “instead of the formula, m...”
(7 votes)
Answer
- Robert Lee
  Posted 9 years ago. Direct link to Robert Lee's post “if you mean find the solu...”
  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.
  Button navigates to signup page
  (18 votes)
Anna
Posted 9 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.
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Huron Tu
  Posted 7 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? : )
  Comment on Huron Tu's post “In 1827, a mathematician ...”
  (9 votes)
blackbean798
Posted a year ago. Direct link to blackbean798's post “Is anyone here in 2022”
Is anyone here in 2022
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(14 votes)
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- Score Creeper
  Posted 2 months ago. Direct link to Score Creeper's post “I am, i just posted a que...”
  I am, i just posted a question
  Button navigates to signup page
  (1 vote)
Nafia Farzana
Posted 6 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 ?
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(5 votes)
Answer
- Estelle Pretorius
  Posted 6 years ago. Direct link to Estelle Pretorius's post “If the coefficient of x^2...”
  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.
  Comment on Estelle Pretorius's post “If the coefficient of x^2...”
  (11 votes)
Andy Peter
Posted 2 months 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 ?
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(6 votes)
Answer
- Kim Seidel
  Posted 2 months ago. Direct link to Kim Seidel's post “Yes, you can use the quad...”
  Yes, you can use the quadratic formula for all quadratic equations.
  Button navigates to signup page
  (6 votes)
Karyn Williams
Posted 6 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.
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- nkfonseka
  Posted 5 years ago. Direct link to nkfonseka's post “Start from the beginning ...”
  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.
  Comment on nkfonseka's post “Start from the beginning ...”
  (20 votes)
Patrick
Posted 7 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?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- MBlackwll
  Posted 7 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
  Button navigates to signup page
  (6 votes)

Algebra (all content)

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