Understanding and Using the Quadratic Formula | Lecture notes Earth science

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Gain more insight into the quadra!c formula and how it is used in quadra!c equa!ons.

Google Classroom Facebook Twi#er Email

The quadra!c formula helps you solve quadra!c equa!ons, 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 quadra!c equa!on like this:

Then the formula will help you find the roots of a quadra!c equa!on, i.e. the

values of where this equa!on is solved.

The quadra!c formula

It may look a li#le scary, but you’ll get used to it quickly!

Prac!ce using the formula now.

Worked example

First we need to iden!fy the values for a, b, and c (the coeﬃcients). First step,

make sure the equa!on is in the format from above, :

is the coeﬃcient in front of , so here (note that can’t equal

-- the is what makes it a quadra!c).

is the coeﬃcient in front of the , so here .

is the constant, or the term without any next to it, so here .

Then we plug , , and into the formula:

solving this looks like:

Therefore or .

What does the solu!on tell us?

The two solu!ons are the x-intercepts of the equa!on, i.e. where the curve

crosses the x-axis. The equa!on looks like:

where the solu!ons to the quadra!c formula, and the intercepts are

and .

Now you can also solve a quadra!c equa!on through factoring, comple!ng

the square, or graphing, so why do we need the formula?

Because some!mes quadra!c equa!ons are a lot harder to solve than that

first example.

Second worked example

Let’s try this for an equa!on that is hard to factor:

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

The formula gives us:

We know you can’t take the square root of a nega!ve number without using

imaginary numbers, so that tells us there’s no real solu!ons to this equa!on.

This means that at no point will , the func!on won’t intercept the x-axis.

We can also see this when graphed on a calculator:

Now you’ve got the basics of the quadra!c formula!

There are many more worked examples in the videos to follow.

Tips when using the quadra!c formula

Be careful that the equa!on is arranged in the right form:

or it won’t work!

Make sure you take the square root of the whole , and that

is the denominator of everything above it

Watch your nega!ves: can’t be nega!ve, so if starts as nega!ve, make

sure it changes to a posi!ve since the square of a nega!ve or a posi!ve is

a posi!ve

Keep the and always be on the look out for TWO solu!ons

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. and

Next step:

Prac!ce using the quadra!c equa!on.

Watch Sal do an example:

Prove the quadra!c formula:

The quadra!c formula

Download Understanding and Using the Quadratic Formula and more Lecture notes Earth science in PDF only on Docsity!

All in for Education

"We'll get right to the point: we're asking you to help. Khan Academy is a nonprofit that relies on support from people like you. If everyone reading this gives $10 monthly, we can con!nue to thrive for years. Gi"s through midnight Dec 31 will be matched. Give now!" -Sal Khan, Founder & CEO Yes, I'll generously add $0.25 each month to cover the transac!on fees so Khan Academy can keep 100% of my dona!on. ▸ Click here to give in honor of another person By dona!ng, you agree to our terms of service and privacy policy. Monthly One !me Select gi" frequency $10 $20 $30 $40 $50 Other Select amount Name Give by credit card Give with PayPal Privacy - Terms

Gain more insight into the quadra!c formula and how it is used in quadra!c equa!ons.

Google Classroom Facebook Twi#er Email

The quadra!c formula helps you solve quadra!c equa!ons, 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 quadra!c equa!on like this:

Then the formula will help you find the roots of a quadra!c equa!on, i.e. the

values of where this equa!on is solved.

The quadra!c formula

It may look a li#le scary, but you’ll get used to it quickly!

Prac!ce using the formula now.

Worked example

First we need to iden!fy the values for a, b, and c (the coefficients). First step,

make sure the equa!on is in the format from above, :

is the coefficient in front of , so here (note that can’t equal

-- the is what makes it a quadra!c).

is the coefficient in front of the , so here.

is the constant, or the term without any next to it, so here.

Then we plug , , and into the formula:

solving this looks like:

Therefore or.

What does the solu!on tell us?

The two solu!ons are the x-intercepts of the equa!on, i.e. where the curve

crosses the x-axis. The equa!on looks like:

where the solu!ons to the quadra!c formula, and the intercepts are

and.

Now you can also solve a quadra!c equa!on through factoring, comple!ng

the square, or graphing, so why do we need the formula?

Because some!mes quadra!c equa!ons are a lot harder to solve than that

first example.

Second worked example

Let’s try this for an equa!on that is hard to factor:

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

The formula gives us:

We know you can’t take the square root of a nega!ve number without using

imaginary numbers, so that tells us there’s no real solu!ons to this equa!on.

This means that at no point will , the func!on won’t intercept the x-axis.

We can also see this when graphed on a calculator:

Now you’ve got the basics of the quadra!c formula!

There are many more worked examples in the videos to follow.

Tips when using the quadra!c formula

Be careful that the equa!on is arranged in the right form:

or it won’t work!

Make sure you take the square root of the whole , and that

is the denominator of everything above it

Watch your nega!ves: can’t be nega!ve, so if starts as nega!ve, make

sure it changes to a posi!ve since the square of a nega!ve or a posi!ve is

a posi!ve

Keep the and always be on the look out for TWO solu!ons

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. and

Next step:

Prac!ce using the quadra!c equa!on.

Watch Sal do an example:

Prove the quadra!c formula:

The quadra!c formula

Understanding the quadra!c formula

ax +

bx + c = 0

x

x =

2 a

− b ± b^ −^4 ac

ax +

bx + c = 0

x +

4 x − 21 = 0

a x

a = 1 a 0

x

b x b = 4

c x c = − 21

a b c

x =

x = 3 x = − 7

x +

3 x − 4 = 0

x = − 4 x = 1

3 x +

6 x = − 10

x +

a

x +

b

c

x =

− 6 ± 36 −^120

y = 0

ax +

bx + c = 0

( b −

4 ac ) 2 a

b

Example: Quadratics in standard form | Quadratic equati… Watch later Share Proof of quadratic formula | Polynomial and rational func… Watch later Share The quadra!c formula Understanding the quadra!c formula Worked example: quadra!c formula (example 2) Worked example: quadra!c formula (nega!ve coefficients) Prac!ce: Quadra!c formula Using the quadra!c formula: number of solu!ons Prac!ce: Number of solu!ons of quadra!c equa!ons Proof of the quadra!c formula Quadra!c formula review Discriminant review Quadra!c formula proof review Next lesson

Comple!ng the square

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Log in See 4 more replies Cian Knight 4 years ago Where does the word "Quadra!c" come from? Reply • 1 comment (43 votes)^ Upvote^ Downvote^ Flag^ more Adithi J 4 years ago Good ques!on! It is derived from the La!n word quadrare, which means "to square", which is what you do in quadra!cs. 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.) 2 comments (80 votes)^ Upvote^ Downvote^ Flag^ more See 5 more replies Carlos Kieliszewski 3 years ago I am just a middle school student who learned this for fun, and I was just wondering: How would I use this in real life? Is this an equa!on for an everyday lifestyle? Thanks! Edit: I just looked back at this a"er a few years, and I'm now in Algebra II in high school. I'm not sure why I thought this was an ordinary day formula but it brings back memories... Reply • 5 comments (17 votes)^ Upvote^ Downvote^ Flag^ more ryancoykendall 3 years ago I am an adult trying to go back to college and using Khan Academy to brush up on my math skills. Knowing why math is important for "the real world" is cri!cal to me. I did not have the benefit of being taught the real world applica!ons of algebra in high school so I mentally checked out and only did what was necessary to pass the tests... As a healthcare provider, I now use algebra occasionally to calculate medica!on drip rates but the quadra!c equa!on, not so much. So I looked it up and there are some really cool applica!ons of the quadra!c formula: such as calcula!ng the arc of a projec!le (and this applies to much more interes!ng things than just throwing a baseball up in the air, tasks like calcula!ng the flight path of a bullet or a missile), or for calcula!ng the resistance of a resistor in an electrical circuit, etc. These are not tasks I perform in my line of work but this equa!on applies to real professions. The quadra!c equa!on is part of a language to describe physical ac!ons in the real world! That is exci!ng to me and mo!vates me to try my best to learn how to speak the language for myself! Check outh#ps://h#p://www.mathsisfun.com/algebra/quadra!c-equa!on-real- world.html 1 comment (29 votes)^ Upvote^ Downvote^ Flag^ more See 4 more replies Daniel Rendall 6 years ago does x2 = x to the power of 2? Reply • 4 comments (5 votes)^ Upvote^ Downvote^ Flag^ more stephen 6 years ago Yes x with a li#le 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 conven!on is to use the "^" symbol to say "to the power of" so x to the power of 2 would be x^ 1 comment (21 votes)^ Upvote^ Downvote^ Flag^ more See 1 more reply Anna 5 years ago Could you extend this quadra!c formula to work for other non-linear equa!ons as well? I mean I have heard of so called Oc!c Equa!ons 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 deriva!ve, or e to mean 2.7... or f, g, and h to mean func!on of x or i to mean the imaginary unit, just as variables. Reply • Comment (5 votes)^ Upvote^ Downvote^ Flag^ more Thomas B 5 years ago There is one for 3rd degree (cubic) equa!ons although it is quite nasty, and an even uglier one for 4th degree (quar!c)) problems that is even more disgus!ng, but I don't believe there are ones beyond that. Cubic - h#p://www.math.vanderbilt.edu/~schectex/courses/cubic/ Quar!c - h#p://planetmath.org/quar!cformula Comment (12 votes)^ Upvote^ Downvote^ Flag^ more See 1 more reply Sam D 3 years ago Just curious, is there something like the "Trinomial formula", for third degree polynomials and so on? Or do we figure it out by normal factoriza!on? So what makes second degree polynomials so special over say, 5th, or 3rd degree ones? Reply • Comment (5 votes)^ Upvote^ Downvote^ Flag^ more andrewp18 3 years ago Good ques!on! 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 quadra!c polynomial. There are general formulas for 3rd degree and 4th degree polynomials as well. These are the cubic and quar!c formulas. Both of these formulas are significantly more complicated and difficult to derive than the 2nd degree quadra!c formula! Here is a picture of the full quar!c formula: h#ps://upload.wikimedia.org/wikipedia/commons/9/99/Quar!c_Formul a.svg Be sure to scroll down and to the right to see the full formula! It's huge! In prac!ce, there are other more efficient methods that we can employ to solve cubics and quar!cs 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 quar!c). 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: h#ps://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. 2 comments (7 votes)^ Upvote^ Downvote^ Flag^ more See 2 more replies kit wing 6 years ago instead of the formula, my textbook wants me to use factoriza!on..how to i do x^2+2x-3=0?

how do i factorize x^2+2-3?
is it possible to use the formula for this? (i tried but cldnt seem to find the answer Reply • 1 comment (2 votes)^ Upvote^ Downvote^ Flag^ more Robert Lee 6 years ago if you mean find the solu!on, yes, you would get -3 and 1. If you want to factor it would be (x + 3) (x - 1). The quadra!c formula helps you find the roots not the factored form. Comment (10 votes)^ Upvote^ Downvote^ Flag^ more farzananafia 2 years ago How do i know when the curve goes like a u or a upside down u? Reply • Comment (3 votes)^ Upvote^ Downvote^ Flag^ more Estelle Pretorius 2 years ago If the coefficient of x^2 is nega!ve, 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 posi!ve, the curve will look like a u (i.e. the curve will have an absolute minimum). Hope this helps. Comment (5 votes)^ Upvote^ Downvote^ Flag^ more Megan Tseng 3 years ago I forgot what's Quadra!c Formula 2? Because I knows there's one for when "b" is not divisible by 2 and one for when "b" is, which is the second one. Reply • Comment (3 votes)^ Upvote^ Downvote^ Flag^ more kubleeka 3 years ago There's only one quadra!c formula, and it works for any value of a, b, or c, not ma#er what they're divisible by (as long as a is not 0). 2 comments (3 votes)^ Upvote^ Downvote^ Flag^ more likhit.krishnam2003 4 years ago what is the purpose of the quadra!c formula? How is it used in a real - life applica!on? Reply • 1 comment (3 votes)^ Upvote^ Downvote^ Flag^ more Dekin O'Sullivan 2 years ago Here is an example of a real life applicaton in physics: h#ps://www.khanacademy.org/science/physics/two-dimensional- mo!on/two-dimensional-projec!le-mot/v/launching-and-landing-on- different-eleva!ons Comment (1 vote)^ Upvote^ Downvote^ Flag^ more See 1 more reply Patrick 4 years ago For the quadra!c formula, I have a quick ques!on. For the b^2 part inside the square root, why can't it be transferred to the outside as b? Reply • Comment (2 votes)^ Upvote^ Downvote^ Flag^ more MBlackwll 4 years ago Hopefully this proof helps you understand why: h#ps://www.khanacademy.org/math/algebra/quadra!cs/solving- quadra!cs-using-the-quadra!c-formula/v/proof-of-quadra!c-formula Comment (3 votes)^ Upvote^ Downvote^ Flag^ more Show more... Ques!ons Tips & Thanks

The quadra!c formula Worked example: quadra!c formula (example 2)

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Understanding and Using the Quadratic Formula, Lecture notes of Earth science

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Download Understanding and Using the Quadratic Formula and more Lecture notes Earth science in PDF only on Docsity!

All in for Education

Google Classroom Facebook Twi#er Email

The quadra!c formula helps you solve quadra!c equa!ons, 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 quadra!c equa!on like this:

Then the formula will help you find the roots of a quadra!c equa!on, i.e. the

values of where this equa!on is solved.

The quadra!c formula

It may look a li#le scary, but you’ll get used to it quickly!

Prac!ce using the formula now.

Worked example

First we need to iden!fy the values for a, b, and c (the coefficients). First step,

make sure the equa!on is in the format from above, :

is the coefficient in front of , so here (note that can’t equal

-- the is what makes it a quadra!c).

is the coefficient in front of the , so here.

is the constant, or the term without any next to it, so here.

Then we plug , , and into the formula:

solving this looks like:

Therefore or.

What does the solu!on tell us?

The two solu!ons are the x-intercepts of the equa!on, i.e. where the curve

crosses the x-axis. The equa!on looks like:

where the solu!ons to the quadra!c formula, and the intercepts are

and.

Now you can also solve a quadra!c equa!on through factoring, comple!ng

the square, or graphing, so why do we need the formula?

Because some!mes quadra!c equa!ons are a lot harder to solve than that

first example.

Second worked example

Let’s try this for an equa!on that is hard to factor:

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

The formula gives us:

We know you can’t take the square root of a nega!ve number without using

imaginary numbers, so that tells us there’s no real solu!ons to this equa!on.

This means that at no point will , the func!on won’t intercept the x-axis.

We can also see this when graphed on a calculator:

Now you’ve got the basics of the quadra!c formula!

There are many more worked examples in the videos to follow.

Tips when using the quadra!c formula

Be careful that the equa!on is arranged in the right form:

or it won’t work!

Make sure you take the square root of the whole , and that

is the denominator of everything above it

Watch your nega!ves: can’t be nega!ve, so if starts as nega!ve, make

sure it changes to a posi!ve since the square of a nega!ve or a posi!ve is

a posi!ve

Keep the and always be on the look out for TWO solu!ons

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. and

Next step:

Prac!ce using the quadra!c equa!on.

Watch Sal do an example:

Prove the quadra!c formula:

Tags

Understanding the quadra!c formula

ax +

bx + c = 0

x

x =

2 a

− b ± b^ −^4 ac

ax +

bx + c = 0

x +

4 x − 21 = 0

a x

a = 1 a 0

x

b x b = 4

c x c = − 21

a b c