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Completing the square among other things

  1. Jun 7, 2004 #1
    I need to know how to complete the square.. every website I go to about this is completely complicated.. my math teacher does it like so:
    y = x^2 + 6x -7
    y = x^2 + 6x + 3^2 - 9 - 7
    y = (x + 3)^2 - 16

    that I understand but..

    if it were to say y = 2x^2 + 6x - 7

    that's where I get confused because of the 2 infront of the x^2

    I appreciate any help... -scaez
     
  2. jcsd
  3. Jun 7, 2004 #2
    Factor it out.

    y = 2(x^2 + 3x + 7/2)

    Leave the 2 out front and complete the square of x^2 + 3x + 7/2.

    cookiemonster
     
  4. Jun 8, 2004 #3

    HallsofIvy

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    (x+a)<sup>2</sup>= x<sup>2</sup>+ 2ax+ a<sup>2</sup>.

    To complete a square such as x<sup>2</sup>+ 3x, compare with that: we must have
    2ax= 3x or a= 3/2. That gives a<sup>2</sup>= 9/4. We complete the square in x<sup>2</sup>+ 3x by adding 9/4: x<sup>2</sup>+ 3x+ 9/4= (x+ 3/2)<sup>2</sup>.

    I would be inclined to complete the square in 2x<sup>2</sup>+ 6x+ 7 by factoring the 2 out of the first two terms only: 2(x<sup>2</sup>+ 3x)+ 7. As before, to complete the square, add and subtract 9/4: 2(x<sup>2</sup>+ 3x+ 9/4- 9/4)+ 7/2. Since I ignored the "2", both 9/4s are inside the parentheses. I'll need to take -9/4 out of the parentheses and multiply by 2:
    2x<sup>2</sup>+ 6x+ 7= 2(x<sup>2</sup>+ 3x+ 9/4- 9/4)+ 7= 2(x<sup>2</sup>+ 3x+ 9/4)- 9/2+ 7= (x+ 3/2)<sup>2</sup>+ 5/2.
     
  5. Jun 8, 2004 #4

    jcsd

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    If yu like you can also put the factor back in the brackets:

    y = 2x^2 + 3x + 7 = 2(x + 3/2)^2 + 5/4 = (sqrt(2)x + 3/sqrt(2))^2 + 5/4
     
  6. Jun 8, 2004 #5
    You start with a polynomial

    ax^2 + bx + c = 0

    You factor “a” out of any terms with x

    [tex] a(x^2 + \frac{b}{a}x) + c = 0 [/tex]

    Then you divide b/a by ½ and square it.

    So [tex] (\frac{b}{2a}})^2 [/tex]

    Then you add that number inside the parenthesizes and subtract it multiplied by “a” (to keep the equation balanced)

    [tex] a(x^2 + \frac{b}{a} x + (\frac{b}{2a})^2 ) + c -a{\frac{b}{2a}}^2 = 0 [/tex]

    For example if you wanted to do the completing the square method for [tex]2x^2 + 16x +17 = -15[/tex]

    First you factor out a
    [tex] 2(x^2 + 8x) +17 = -15[/tex]

    then divide your b/a term by ½ and square it
    (8/2)^2 = 16

    add it to the inside and subtract it from the outside (multiply the outside one by “a”)

    [tex] 2(x^2 +8x + 16) - 32 + 17 = -15 [/tex]

    factor and simplify

    [tex] 2(x+4)(x+4) = 0 [/tex]

    x=-4
     
  7. Jun 12, 2004 #6
    Here's another, maybe simpler way to look at it:

    Starting with the general quadratic equation: (note - I lost my superscripts in translation here - I hope this doesn't confuse you!)

    ax2 + bx+ c = 0

    Subtract c from both sides:

    ax2 + bx = -c

    Divide both sides by a (we can do this since we know a does not equal 0. If it did, we wouldn’t have a quadratic equation)

    x2 + (b/a)x = - c/a

    Now comes the part where most people get confused. But, for now we’ll just do this. I’ll explain it in a second. Divide b/a by 2, and add the square of that to the right hand side of the equation. Then, change the left hand side of the equation so that it looks like this:

    (x +b/2a)2 = b2/4a2 – c/a

    Ok, all this comes from examining what happens when we square a binomial:
    Consider:

    (a + d) = 0

    Squaring both sides:

    (a+d)2 = 0

    Which gives us: (Use the FOIL method – First, Outer, Inner, Last)

    a2 + 2ad + d2 = 0

    Notice the factor of 2 in the middle term on the left. So, if we want the middle term in our original equation (x2 + (b/a)x = -c/a) to come out right, the second term in the binomial we want to square has to be b/2a; half of b/a Now also notice that if we square (x+ b/2a) we will get a final term of b2/4a2 which didn’t appear in our original equation. But that’s easily fixed – just add it to the right hand side also.

    So, moving on, we can add the two fractions on the right hand side of our new equation and get:

    (x + b/2a)2 = (b2 – 4ac)/2a

    And you have completed the square!

    Note, that if you solve this for x you get the famous general solution for the quadratic equation.
     
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