Sal, How does the quadratic formula relate to business and economics?

It helps in lots of ways. It can possibly predict the future path of certain things, especially if your graph is exponential.

When there are two numbers in the numerator before the square root, how do I solve the problem?

Can you show an example of what you mean?

are there any shortcuts or patterns we can use to make calculation quicker?

Not insofar as I know. The quadratic formula is the shortcut, unless you prefer grouping or something

what if the equation doesn't equal zero

then just subtract the non-zero number from the RHS to the LHS and make the RHS equal to zero.

isn't it the square root of -112?

If you are referring to the practice problem, use the "explain" link to see how to derive the solution. Compare the work in the explanation to your work to find your error.

I do not understand this. Can someone please explain

This is a formula, so if you can get the right numbers, you plug them into the formula and calculate the answer(s). We always have to start with a quadratic in standard form: ax^2+bx+c=0. Making one up, 3x^2+2x-5=0, we see a=3, b=2, c=-5. I teach my students to start with the discriminant, b^2-4ac. Also, especially in the beginning, put the b value in parentheses so that you square a negative number if b is negative. In our example, this gives (2)^2-4(3)(-5) = 4+60=64. If I take √64 = 8. Filling out the formula, we get x=(-2±8)/(2(3)) or breaking it into the two parts x=(-2+8)/6=1 and (-2-8)/6=-10/6=-5/2. Where is the confusion? It is always hard to answer when we cannot figure out what you do understand and where you are confused.

can you reccomend other math websites for algebra 1 and 2

IXL is a good one, but without a login given to you by an administrator, you have to pay a membership fee.

What happens when the discriminant is a negative number? If it is negative would your answer be imaginary?

Yes... If the discriminant is negative, then there are 2 roots, but they are complex numbers.

Main content

Algebra 1

Course: Algebra 1 > Unit 14

Lesson 6: The quadratic formula

Quadratic formula review

Google Classroom

The quadratic formula allows us to solve any quadratic equation that's in the form ax^2 + bx + c = 0. This article reviews how to apply the formula.

What is the quadratic formula?

The quadratic formula says that

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

for any quadratic equation like:

a x^{2} + b x + c = 0

Example

We're given an equation and asked to solve for

q

0 = - 7 q^{2} + 2 q + 9

This equation is already in the form

a x^{2} + b x + c = 0

, so we can apply the quadratic formula where

a = - 7, b = 2, c = 9

\begin{aligned} q & = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ q & = \frac{- 2 \pm \sqrt{2^{2} - 4 (- 7) (9)}}{2 (- 7)} \\ q & = \frac{- 2 \pm \sqrt{4 + 252}}{- 14} \\ q & = \frac{- 2 \pm \sqrt{256}}{- 14} \\ q & = \frac{- 2 \pm 16}{- 14} \\ q & = \frac{- 2 + 16}{- 14}, q = \frac{- 2 - 16}{- 14} \\ q & = - 1, q = \frac{9}{7} \end{aligned}

Let's check both solutions to be sure it worked:

$q = - 1$ ‍	$q = \frac{9}{7}$ ‍
$\begin{aligned} 0 & = - 7 q^{2} + 2 q + 9 \\ 0 & = - 7 (- 1)^{2} + 2 (- 1) + 9 \\ 0 & = - 7 (1) - 2 + 9 \\ 0 & = - 7 - 2 + 9 \\ 0 & = 0 \end{aligned}$ ‍	$\begin{aligned} 0 & = - 7 q^{2} + 2 q + 9 \\ 0 & = - 7 {(\frac{9}{7})}^{2} + 2 (\frac{9}{7}) + 9 \\ 0 & = - 7 (\frac{81}{49}) + (\frac{18}{7}) + 9 \\ 0 & = - (\frac{81}{7}) + (\frac{18}{7}) + 9 \\ 0 & = - (\frac{63}{7}) + 9 \\ 0 & = - 9 + 9 \\ 0 & = 0 \end{aligned}$ ‍

Yep, both solutions check out.

Want to learn more about the quadratic formula? Check out this video.

Practice

Solve for $x$ ‍.

- 4 + x + 7 x^{2} = 0

Want more practice? Check out this exercise.

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Charlie Williams
Posted 4 years ago. Direct link to Charlie Williams's post “Sal, How does the quadrat...”
Sal, How does the quadratic formula relate to business and economics?
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- Victor
  Posted 4 years ago. Direct link to Victor's post “It helps in lots of ways....”
  It helps in lots of ways. It can possibly predict the future path of certain things, especially if your graph is exponential.
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kaycoach03
Posted 4 years ago. Direct link to kaycoach03's post “what if the equation does...”
what if the equation doesn't equal zero
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- amy
  Posted 4 years ago. Direct link to amy's post “then just subtract the no...”
  then just subtract the non-zero number from the RHS to the LHS and make the RHS equal to zero.
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rylan.wetsell
Posted a year ago. Direct link to rylan.wetsell's post “are there any shortcuts o...”
are there any shortcuts or patterns we can use to make calculation quicker?
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- Emma Lai
  Posted 8 months ago. Direct link to Emma Lai's post “Not insofar as I know. Th...”
  Not insofar as I know. The quadratic formula is the shortcut, unless you prefer grouping or something
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  (5 votes)
trentonbothwell2015
Posted 5 months ago. Direct link to trentonbothwell2015's post “When there are two number...”
When there are two numbers in the numerator before the square root, how do I solve the problem?
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- David Severin
  Posted 5 months ago. Direct link to David Severin's post “Can you show an example o...”
  Can you show an example of what you mean?
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Tiggertast
Posted 4 years ago. Direct link to Tiggertast's post “can you reccomend other m...”
can you reccomend other math websites for algebra 1 and 2
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mdondapati29
Posted 10 months ago. Direct link to mdondapati29's post “isn't it the square root ...”
isn't it the square root of -112?
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- Kim Seidel
  Posted 10 months ago. Direct link to Kim Seidel's post “If you are referring to t...”
  If you are referring to the practice problem, use the "explain" link to see how to derive the solution. Compare the work in the explanation to your work to find your error.
  Button navigates to signup page
  (2 votes)
Wolfplanet
Posted a year ago. Direct link to Wolfplanet's post “I do not understand this....”
I do not understand this. Can someone please explain
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(1 vote)
Answer
- David Severin
  Posted a year ago. Direct link to David Severin's post “This is a formula, so if ...”
  This is a formula, so if you can get the right numbers, you plug them into the formula and calculate the answer(s). We always have to start with a quadratic in standard form: ax^2+bx+c=0. Making one up, 3x^2+2x-5=0, we see a=3, b=2, c=-5. I teach my students to start with the discriminant, b^2-4ac. Also, especially in the beginning, put the b value in parentheses so that you square a negative number if b is negative. In our example, this gives (2)^2-4(3)(-5) = 4+60=64. If I take √64 = 8. Filling out the formula, we get x=(-2±8)/(2(3)) or breaking it into the two parts x=(-2+8)/6=1 and (-2-8)/6=-10/6=-5/2. Where is the confusion? It is always hard to answer when we cannot figure out what you do understand and where you are confused.
  Comment on David Severin's post “This is a formula, so if ...”
  (4 votes)
Kelvonna Hicks
Posted 3 years ago. Direct link to Kelvonna Hicks's post “can you reccomend other m...”
can you reccomend other math websites for algebra 1 and 2
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(1 vote)
Answer
- Mackenzy Combs
  Posted 3 years ago. Direct link to Mackenzy Combs's post “IXL is a good one, but wi...”
  IXL is a good one, but without a login given to you by an administrator, you have to pay a membership fee.
  Button navigates to signup page
  (3 votes)
Drake
Posted 5 years ago. Direct link to Drake's post “What happens when the dis...”
What happens when the discriminant is a negative number? If it is negative would your answer be imaginary?
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(1 vote)
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- Kim Seidel
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  Yes... If the discriminant is negative, then there are 2 roots, but they are complex numbers.
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  (3 votes)
JAKAYLINS
Posted 9 months ago. Direct link to JAKAYLINS's post “Can i have exra work.”
Can i have exra work.
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