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.

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.

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.

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.

How do you get from (-4 +-2 sqrt of 34)/-6 to (-2+- sqrt of 34)/-3 ? I get that it’s divided by 2 but it’s strange that the square root of 34 doesn’t change.

You can only remove the factor of 2 once from each term. There are two terms: -4 and +/- 2 sqrt(34) -4/2 = -2 +/- 2 sqrt(34)/2 = +/- sqrt(34) Also, the 2 contained in the sqrt(34) is sqrt(2) which can't be divided by 2. They are not equal values. Hope this helps.

Main content

Algebra 1

Course: Algebra 1 > Unit 14

Lesson 6: The quadratic formula

Quadratic formula review

CCSS.Math:

HSA.REI.B.4

HSA.REI.B.4b

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, equals, start fraction, minus, start color #e07d10, b, end color #e07d10, plus minus, square root of, start color #e07d10, b, end color #e07d10, squared, minus, 4, start color #7854ab, a, end color #7854ab, start color #e84d39, c, end color #e84d39, end square root, divided by, 2, start color #7854ab, a, end color #7854ab, end fraction

for any quadratic equation like:

start color #7854ab, a, end color #7854ab, x, squared, plus, start color #e07d10, b, end color #e07d10, x, plus, start color #e84d39, c, end color #e84d39, equals, 0

Example

We're given an equation and asked to solve for q:

0, equals, minus, 7, q, squared, plus, 2, q, plus, 9

This equation is already in the form a, x, squared, plus, b, x, plus, c, equals, 0, so we can apply the quadratic formula where a, equals, minus, 7, comma, b, equals, 2, comma, c, equals, 9:

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

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

q, equals, minus, 1	q, equals, start fraction, 9, divided by, 7, end fraction
$\begin{aligned}0&=-7q^2+2q+9\\\\0&=-7(-1)^2+2(-1)+9 \\\\0&=-7(1)-2+9 \\\\0&=-7-2+9\\\\0&=0\end{aligned}$	$\begin{aligned}0&=-7q^2+2q+9\\\\0&=-7\left(\dfrac{9}{7}\right)^2+2\left (\dfrac{9}{7}\right)+9 \\\\0&=-7\left(\dfrac{81}{49}\right)+\left (\dfrac{18}{7}\right)+9 \\\\0&=-\left(\dfrac{81}{7}\right)+\left (\dfrac{18}{7}\right)+9 \\\\0&=-\left(\dfrac{63}{7}\right) +9 \\\\0&=-9 +9 \\\\0&=0\end{aligned}$

Yep, both solutions check out.

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

Practice

Solve for x.

minus, 4, plus, x, plus, 7, x, squared, equals, 0

(Choice A)
x, equals, start fraction, minus, 7, plus minus, square root of, 65, end square root, divided by, 8, end fraction
(Choice B)
x, equals, start fraction, minus, 1, plus minus, square root of, 5, end square root, divided by, 2, end fraction
(Choice C)
x, equals, start fraction, minus, 1, plus minus, square root of, 113, end square root, divided by, 14, end fraction
(Choice D)
x, equals, start fraction, 3, plus minus, square root of, 21, end square root, divided by, 6, end fraction

Want more practice? Check out this exercise.

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Charlie Williams
Posted 3 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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(7 votes)
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- Victor
  Posted 3 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.
  Comment on Victor's post “It helps in lots of ways....”
  (16 votes)
kaycoach03
Posted 3 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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(3 votes)
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- amy
  Posted 3 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.
  Comment on amy's post “then just subtract the no...”
  (10 votes)
Tiggertast
Posted 3 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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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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- Kim Seidel
  Posted 5 years ago. Direct link to Kim Seidel's post “Yes... If the discriminan...”
  Yes... If the discriminant is negative, then there are 2 roots, but they are complex numbers.
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Kelvonna Hicks
Posted 2 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 2 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
  (2 votes)
louisaandgreta
Posted a year ago. Direct link to louisaandgreta's post “How do you get from (-4 +...”
How do you get from (-4 +-2 sqrt of 34)/-6

to (-2+- sqrt of 34)/-3 ?

I get that it’s divided by 2 but it’s strange that the square root of 34 doesn’t change.
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “You can only remove the f...”
  You can only remove the factor of 2 once from each term. There are two terms: -4 and +/- 2 sqrt(34)
  -4/2 = -2
  +/- 2 sqrt(34)/2 = +/- sqrt(34)
  
  Also, the 2 contained in the sqrt(34) is sqrt(2) which can't be divided by 2. They are not equal values.
  
  Hope this helps.
  Comment on Kim Seidel's post “You can only remove the f...”
  (2 votes)
Anaya DeLeon
Posted 3 years ago. Direct link to Anaya DeLeon's post “A garden in the shape of ...”
A garden in the shape of a rectangle is surrounded by a walkway of uniform width. The dimensions of the garden only are 35 by 24. The area of the garden and the walkway together is 1,530 square feet. What is the width of the walkway in feet? a. 4 ft b. 5 ft c. 34.5 ft
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Answer
- samsajwani1234
  Posted a year ago. Direct link to samsajwani1234's post “Assume the length of the ...”
  Assume the length of the walkway to be x. If you illustrate the data as a drawing on a piece of paper, you will see that the sides of the garden + walkway are (35+2x) and (24 + 2x). Since AREA = LENGTH x WIDTH, we get to the equation:
  (35 + 2x) (24 + 2x) = 1530
  
  By expanding the L.H.S, we get
  840 + 70x + 48x + 4x^2 = 1530
  
  Now after rearranging
  4x^2 + 118x + 840 - 1530 = 0 -----> 4x^2 + 118x - 690 = 0
  
  After applying quadratic formula (a = 4, b = 118, c = -690) we get
  x = 5 ft OR x = -34.5 ft
  
  Since the length can NEVER be negative, we can exclude x = -34.5 ft. *SO THE ANSWER IS b.5ft*
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  (1 vote)
rylan.wetsell
Posted 3 months 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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Wolfplanet
Posted 9 months 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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- David Severin
  Posted 9 months 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 ...”
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lmbaldwin
Posted 4 years ago. Direct link to lmbaldwin's post “What if there was a negat...”
What if there was a negative number under the square root? What would you do then?
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- Kim Seidel
  Posted 4 years ago. Direct link to Kim Seidel's post “If there is a negative nu...”
  If there is a negative number inside the radical, then the roots will be complex numbers. See this link: https://www.khanacademy.org/math/algebra2/polynomial-functions/quadratic-equations-with-complex-numbers/v/complex-roots-from-the-quadratic-formula
  
  If you haven't learned about "i", the imaginary number, then you should start at this link: https://www.khanacademy.org/math/algebra2/introduction-to-complex-numbers-algebra-2/the-imaginary-numbers-algebra-2/v/introduction-to-i-and-imaginary-numbers
  Comment on Kim Seidel's post “If there is a negative nu...”
  (1 vote)