Quadratic Equation
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The Quadratic Formula Explained
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The Quadratic Formula Explained (page 1 of 3)
Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution. The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process of completing the square, and is formally stated as: For ax2 +
bx + c = 0, the value of x is given by:
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For the Quadratic Formula to work, you must have your equation arranged in the form "(quadratic) = 0". Also, the "2a" in the denominator of the Formula is underneath everything above, not just the square root. And it's a "2a" under there, not just a plain "2". Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget to "put them back" on your test, and you'll mess yourself up. Remember that "b2" means "the square of ALL of b, including its sign", so don't leave b2 being negative, even if b is negative, because the square of a negative is a positive. In other words, don't be sloppy and don't try to take shortcuts, because it will only hurt you in the long run. Trust me on this!
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Here are some examples of how the Quadratic Formula works: Solve x2 +
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3x – 4 = 0
This quadratic happens to factor:
x2 + 3x – 4 = (x + 4)(x – 1) = 0
...so I already know that the solutions are x = –4 and x = 1. How would my solution look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, my solution looks like this:
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Then, as expected, the solution is x =
–4, x = 1.
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The Quadratic Formula Explained
Page 2 of 3
+ bx + c = y, and you are told to plug zero in for y. The corresponding x+ bx + c = 0 for x means, among other things, that you are trying to find x-intercepts. Since there were two solutions for x2 + 3x – 4 = 0, there must then be two x-intercepts on the graph. Graphing, we get the curve below:
values are the x-intercepts of the graph. So solving ax2
Suppose you have ax2
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As you can see, the x-intercepts match the solutions, crossing the x-axis at x = –4 and x = 1. This shows the connection between graphing and solving: When you are solving "(quadratic) = 0", you are finding the x-intercepts of the graph. This can be useful if you have a graphing calculator, because you can use the Quadratic Formula (when necessary) to solve a quadratic, and then use your graphing calculator to make sure that the displayed x-intercepts have the same decimal values as do the solutions provided by the Quadratic Formula. Note, however, that the calculator's display of the graph will probably have some pixel-related roundoff error, so you'd be checking to see that the computed and graphed values were reasonably close; don't expect an exact match. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved Solve 2x2 –
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4x – 3 = 0. Round your answer to two decimal places, if necessary.
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There are no factors of (2)(–3) = –6 that add up to –4, so I know that this quadratic cannot be factored. I will apply the Quadratic Formula. In this case, a = 2, b = –4, and c = –3:
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Then the answer is x
= –0.58, x = 2.58, rounded to two decimal places.
Warning: The "solution" or "roots" or "zeroes" of a quadratic are usually required to be in the "exact" form of the answer. In the example above, the exact form is the one with the square roots of ten in it. You'll need to get a calculator approximation in order to graph the x-intercepts or to simplify the final answer in a word problem. But unless you have a good reason to think that the answer is supposed to be a rounded answer, always go with the exact form.
Compare the solutions of 2x2 intercepts of the graph:
– 4x – 3 = 0 with the x-
Remember: The "solutions" of an equation are also the x-
http://www.purplemath.com/modules/quadform.htm
23/06/2011
The Quadratic Formula Explained
Page 3 of 3
intercepts of the corresponding graph.
Top | 1 | 2 | 3 | Return to Index Next >>
Cite this article as: Stapel, Elizabeth. "The Quadratic Formula Explained." Purplemath. Available from http://www.purplemath.com/modules/quadform.htm. Accessed 23 June 2011 Feedback | Error?
Copyright © 2000-2011 Elizabeth Stapel | About | Terms of Use
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Page 1 of 3
IBM Power Systems See How IBM Can Lower TCO & Raise Confidence In Your Infrastructure. Exercise Your Brain Games You Didn't Know Existed to Fight Brain Decline and Aging. Scholarship Exams Prep All levels. All areas. Online Info Pack. Test Your Skills!
IBM.com/AU/IBM_Power_Servers
www.lumosity.com
www.EdWorksGlobal.com
The Purplemath Forums
Helping students gain understanding and self-confidence in algebra
Search
powered by FreeFind
Return to the Lessons Index | Do the Lessons in Order | Get "Purplemath on CD" for offline use | Print-friendly page
The Quadratic Formula Explained (page 1 of 3)
Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution. The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process of completing the square, and is formally stated as: For ax2 +
bx + c = 0, the value of x is given by:
IBM Servers
Specifically Designed For Small & Medium Businesses. Learn More.
IBM.com/AU/IBM_Servers
For the Quadratic Formula to work, you must have your equation arranged in the form "(quadratic) = 0". Also, the "2a" in the denominator of the Formula is underneath everything above, not just the square root. And it's a "2a" under there, not just a plain "2". Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget to "put them back" on your test, and you'll mess yourself up. Remember that "b2" means "the square of ALL of b, including its sign", so don't leave b2 being negative, even if b is negative, because the square of a negative is a positive. In other words, don't be sloppy and don't try to take shortcuts, because it will only hurt you in the long run. Trust me on this!
Online Video Tutorials
Maths, English Literacy & Grammar Watch, Learn, Extraordinary Results
www.myeducationcompany.…
Here are some examples of how the Quadratic Formula works: Solve x2 +
Brain Training Games
Improve memory with scientifically designed brain exercises.
www.lumosity.com
3x – 4 = 0
This quadratic happens to factor:
x2 + 3x – 4 = (x + 4)(x – 1) = 0
...so I already know that the solutions are x = –4 and x = 1. How would my solution look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, my solution looks like this:
Maths Methods Tutoring
Units 3 & 4 online video lessons. Effective, Convenient & Affordable
www.bequesteducation.com…
$75 Free In Advertising
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Then, as expected, the solution is x =
–4, x = 1.
Purplemath: Linking to this site Printing pages Donating School licensing
Reviews of
http://www.purplemath.com/modules/quadform.htm
23/06/2011
The Quadratic Formula Explained
Page 2 of 3
+ bx + c = y, and you are told to plug zero in for y. The corresponding x+ bx + c = 0 for x means, among other things, that you are trying to find x-intercepts. Since there were two solutions for x2 + 3x – 4 = 0, there must then be two x-intercepts on the graph. Graphing, we get the curve below:
values are the x-intercepts of the graph. So solving ax2
Suppose you have ax2
Internet Sites: Free Help Practice Et Cetera The "Homework Guidelines" Study Skills Survey Tutoring ($$)
This lesson may be printed out for your personal use.
Telstra Mobile PLUS Plans
As you can see, the x-intercepts match the solutions, crossing the x-axis at x = –4 and x = 1. This shows the connection between graphing and solving: When you are solving "(quadratic) = 0", you are finding the x-intercepts of the graph. This can be useful if you have a graphing calculator, because you can use the Quadratic Formula (when necessary) to solve a quadratic, and then use your graphing calculator to make sure that the displayed x-intercepts have the same decimal values as do the solutions provided by the Quadratic Formula. Note, however, that the calculator's display of the graph will probably have some pixel-related roundoff error, so you'd be checking to see that the computed and graphed values were reasonably close; don't expect an exact match. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved Solve 2x2 –
More Flexibility with Data Amounts, Email Solutions & New Bonus Options
www.TelstraBusiness.com/…
Beyond the Balance Sheet
With $27 Billion in Business Assets We Know Tomorrow Makes Today.
www.GECapital.com.au
4x – 3 = 0. Round your answer to two decimal places, if necessary.
Used Ibm Storage
Buy IBM Storage with warranty and available at huge savings off list
www.touchpoint.com.au
There are no factors of (2)(–3) = –6 that add up to –4, so I know that this quadratic cannot be factored. I will apply the Quadratic Formula. In this case, a = 2, b = –4, and c = –3:
Wipe Tests
Measure for possible radioactive contamination
www.tracerco.com.au
Maths Tutoring Solutions
Which solution for prep to year 12 Math Curriculum based Tutor program
www.SchoolWiz.com.au
Then the answer is x
= –0.58, x = 2.58, rounded to two decimal places.
Warning: The "solution" or "roots" or "zeroes" of a quadratic are usually required to be in the "exact" form of the answer. In the example above, the exact form is the one with the square roots of ten in it. You'll need to get a calculator approximation in order to graph the x-intercepts or to simplify the final answer in a word problem. But unless you have a good reason to think that the answer is supposed to be a rounded answer, always go with the exact form.
Compare the solutions of 2x2 intercepts of the graph:
– 4x – 3 = 0 with the x-
Remember: The "solutions" of an equation are also the x-
http://www.purplemath.com/modules/quadform.htm
23/06/2011
The Quadratic Formula Explained
Page 3 of 3
intercepts of the corresponding graph.
Top | 1 | 2 | 3 | Return to Index Next >>
Cite this article as: Stapel, Elizabeth. "The Quadratic Formula Explained." Purplemath. Available from http://www.purplemath.com/modules/quadform.htm. Accessed 23 June 2011 Feedback | Error?
Copyright © 2000-2011 Elizabeth Stapel | About | Terms of Use
Ads by Google
Problem Solving
Quadratic Formula
Quadratic Equation
Graphing Equations
Algebra Equations
http://www.purplemath.com/modules/quadform.htm
23/06/2011
