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Completing the Square:
    Solving Quadratic Equations
(page 1 of 2)


Some quadratics are fairly simple to solve because they are of the form "something-with-x squared equals some number", and then you take the square root of both sides.

An example would be:

       (x – 4)2 = 5
      
    x – 4 = ± sqrt(5)
      
    x = 4 ± sqrt(5)
      
    x = 4 – sqrt(5)  and  x = 4 + sqrt(5)

Unfortunately, most quadratics don't come neatly squared like this. For your average everyday quadratic, you first have to use the technique of "completing the square" to rearrange the quadratic into the neat "(squared part) equals (a number)" format demonstrated above. For example:

  • Find the x-intercepts of y = 4x2 – 2x – 5.

    First off, remember that finding the x-intercepts means setting y equal to zero and solving for the x-values, so this question is really asking you to "Solve 4x2 – 2x – 5 = 0".

This is the original problem. 4x2 – 2x – 5 = 0
Move the loose number over to the other side. 4x2 – 2x = 5
Divide through by whatever is multiplied on the squared term.

Take half of the coefficient (don't forget the sign!) of the x-term, and square it. Add this square to both sides of the equation.

Convert the left-hand side to squared form, and simplify the right-hand side. (This is where you use that sign that you kept track of earlier. You plug it into the middle of the parenthetical part.)

(x – 1/4)^2 = 21/16
Square-root both sides, remembering the "±" on the right-hand side.  Simplify as necessary. x – 1/4 = ± sqrt(21)/4
Solve for "x =". x = 1/4 ± sqrt(21)/4
Remember that the "±" means that you have two values for x. x = 1/4 – sqrt(21)/4 and x = 1/4 + sqrt(21)/4

The answer can also be written in rounded form as

 

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You will need rounded form for "real life" answers to word problems, and for graphing. But (warning!) in most other cases, you should assume that the answer should be in "exact" form, complete with all the square roots.

When you complete the square, make sure that you are careful with the sign on the x-term when you multiply by one-half. If you lose that sign, you can get the wrong answer in the end, because you'll forget what goes inside the parentheses. Also, don't be sloppy and wait to do the plus/minus sign until the very end. On your tests, you won't have the answers in the back, and you will likely forget to put the plus/minus into the answer. Besides, there's no reason to go ticking off your instructor by doing something wrong when it's so simple to do it right. On the same note, make sure you draw in the square root sign, as necessary, when you square root both sides. Don't wait until the answer in the back of the book "reminds" you that you "meant" to put the square root symbol in there. If you get in the habit of being sloppy, you'll only hurt yourself!


  • Solve x2 + 6x – 7 = 0 by completing the square.

    Do the same procedure as above, in exactly the same order. (Study tip: Always working these problems in exactly the same way will help you remember the steps when you're taking your tests.) Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

This is the original equation. x2 + 6x – 7 = 0
Move the loose number over to the other side. x2 + 6x      = 7
Take half of the x-term (that is, divide it by two) (and don't forget the sign!), and square it. Add this square to both sides of the equation. completing-the-square animation
Convert the left-hand side to squared form.  Simplify the right-hand side. (x + 3)2 = 16
Square-root both sides. Remember to do "±" on the right-hand side. x + 3 = ± 4
Solve for "x =". Remember that the "±" gives you two solutions. Simplify as necessary.

   x = – 3 ± 4
      = – 3 – 4, –3 + 4
      = –7, +1

If you are not consistent with remembering to put your plus/minus in as soon as you square-root both sides, then this is an example of the type of exercise where you'll get yourself in trouble. You'll write your answer as "x = –3 + 4 = 1", and have no idea how they got "x = –7", because you won't have a square root symbol "reminding" you that you "meant" to put the plus/minus in. That is, if you're sloppy, these easier problems will embarrass you!

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Cite this article as:

Stapel, Elizabeth. "Completing the Square: Solving Quadratic Equations." Purplemath. Available from
    http://www.purplemath.com/modules/sqrquad.htm. Accessed
 

 

 

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