Completing the Square

Key Questions

  • How do I complete the square if the coefficient of #x^2# is not 1?

    Remember that

    #(ax + b)^2 = a^2x^2 + 2abx + b^2#

    So the general idea to get #b^2# is by getting #x#'s coefficient,
    dividing by #2a#, and squaring the result.

    Example

    #3x^2 + 12x#

    #a = 3#
    #2ab = 12#
    #6b = 12#
    #b = 2#
    #3x^2 + 12x + 4 = (3x + 2)^2#

    You can also factor out #x^2#'s coefficient.. and proceed with completing the square.

    Example:
    #2X^2 +4X#

    #2(X^2 + 2X)#

    #2(X + 1)^2#

    seph · · Oct 7 2014
  • How do I complete the square?

    Answer:

    #ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))#

    The secret is that #b/(2a)# bit

    Explanation:

    Suppose you are given a quadratic equation to solve:

    #2x^2-3x-2 = 0#

    ..which is in the form..

    #ax^2+bx+c = 0# with #a = 2#, #b=-3# and #c=-2#

    #b/(2a) = -3/4#

    So we find:

    #2(x-3/4)^2 = 2(x^2-(2*x*3/4)+(3/4)^2)#

    #=2(x^2-(3x)/2+9/16)#

    #=2x^2-3x+9/8#

    So:

    #2(x-3/4)^2-25/8 = 2(x-3/4)^2-9/8-2#

    #=2x^2-3x+9/8-9/8-2#

    #=2x^2-3x-2#

    So:

    #2x^2-3x-2 = 0#

    turns into:

    #2(x-3/4)^2-25/8 = 0#

    Hence:

    #(x-3/4)^2 = 25/16#

    So:

    #x-3/4 = +-sqrt(25/16) = +-5/4#

    and

    #x = 3/4+-5/4#

    George C. · 2 · Oct 23 2015
  • Does completing the square always work?

    Answer:

    In order to use the Completing the Square method, the value for #a# in the quadratic equation must be #1#. If it is not #1#, you will have to use the AC method or the quadratic formula in order to solve for #x#.

    Explanation:

    Completing the square is a method used to solve a quadratic equation, #ax^2+bx+c#, where #a# must be #1#. The goal is to force a perfect square trinomial on one side and then solving for #x# by taking the square root of both sides.

    The method is explained at the following website:

    http://www.regentsprep.org/regents/math/algtrig/ate12/completesqlesson.htm

    Meave60 · 5 · Nov 30 2015
  • What does completing the square mean?

    Completing the square is a method that represents a quadratic equation as a combination of quadrilateral used to form a square.

    The basis of this method is to discover a special value that when added to both sides of the quadratic that will create a perfect square trinomial.

    That special value is found by evaluation the expression #(b/2)^2# where #b# is found in #ax^2+bx+c=0#. Also in this explanation I assume that #a# has a value of #1#.

    #ax^2+bx+(b/2)^2=(b/2)^2-c#

    #(ax+b/2)^2=(b/2)^2-c#

    #sqrt((ax+b/2)^2)=+-sqrt((b/2)^2-c)#

    #ax+b/2=+-sqrt((b/2)^2-c)#

    #ax=+-sqrt((b/2)^2-c)-b/2#

    A quadratic that is a perfect square is very easy to solve.

    Please take a look at the video to see an example of completing the square visually.

    Completing the Square

    AJ Speller · 1 · Sep 16 2014

Questions

Linear and Quadratic Functions