Results 1 to 14 of 14
Like Tree1Thanks
  • 1 Post By HallsofIvy

Math Help - quadratic equation

  1. #1
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    quadratic equation

    HI;
    I need to complete the square for 9x^2 + 12x + 4.
    but I can't factor out the 9.

    therefore how do I start?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,771
    Thanks
    1140

    Re: quadratic equation

    Quote Originally Posted by anthonye View Post
    HI;
    I need to complete the square for 9x^2 + 12x + 4.
    but I can't factor out the 9.

    therefore how do I start?
    no one said coefficients have to be integers

    $$\begin{align*}&9x^2+12x+4= \\ \\ &9\left(x^2+\frac{4}{3}x+\frac{4}{9}\right)= \\ \\
    &9\left(x+\frac{2}{3}\right)^2= \\ \\ &\left(3x+2\right)^2 \end{align*}$$

    take your pick of the last two forms
    Last edited by romsek; February 18th 2014 at 02:24 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    Ok thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    I just need confirmation that the following steps are correct?

    9x2 +12x + 4 = 0
    9x2 + 12x = - 4
    9(x2 + 4/3 + 4/9) = - 4
    9(x + 2/3)(x + 2/3) = 0

    9(x + 2/3)2 = 0
    (x + 2/3) = 0
    X = + or – 2/3
    = 0.66667
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,771
    Thanks
    1140

    Re: quadratic equation

    Quote Originally Posted by anthonye View Post
    I just need confirmation that the following steps are correct?

    9x2 +12x + 4 = 0
    9x2 + 12x = - 4
    9(x2 + 4/3 + 4/9) = - 4
    9(x + 2/3)(x + 2/3) = 0

    9(x + 2/3)2 = 0
    (x + 2/3) = 0
    X = + or 2/3
    = 0.66667
    No.

    $$9x^2+12x+4=0$$

    $$9\left(x^2+\frac{12}{9}+\frac{4}{9}\right)=0$$

    $$9\left(x^2+\frac{4}{3}+\frac{4}{9}\right)=0$$

    $$9\left(\left(x+\frac{2}{3}\right)^2\right)=0$$

    $$\left(x+\frac{2}{3}\right)^2=0$$

    $$x=-\frac{2}{3}$$
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    how about now?

    9x^2 +12x + 4 = 0
    9x^2 + 12x = - 4
    9(x^2 + 4/3x + 4/9) = - 4
    9(x + 2/3)(x + 2/3) = 0

    9(x + 2/3)^2 = 0
    (x + 2/3) = 0
    X = + or – 2/3
    = 0.66667
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,232
    Thanks
    1795

    Re: quadratic equation

    Anthonye, did you not notice from Romsek's first post that you do NOT "need to complete the square"? $ 9x^2+ 12x+4= (3x+2)^2 $ is already a perfect square!
    Last edited by HallsofIvy; February 19th 2014 at 11:24 AM.
    Thanks from romsek
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    Thanks I didn't notice that.also mostly I just need to know if the process is correct?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    Also how can you tell if a quadratic is a perfect square?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,232
    Thanks
    1795

    Re: quadratic equation

    You start by learning what a "perfect square" is! For any numbers, a and b, $(a+ b)^2=a^2+ 2ab+ b^2 $. The quadratic here is $9x^2+ 12x+ 4 $. Clearly, $a^2= x^2 $ and $b^2= 4$ so a= 3x and b= 2. From that 2ab= 2(3x)(2)= 12x. The whole point of "completing the square" is adding and subtracting that "2ab" part. I don't see how you can learn "completing the square" without first knowing what you want to get.
    Last edited by HallsofIvy; February 19th 2014 at 11:51 AM.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    ok got it thank you both.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,771
    Thanks
    1140

    Re: quadratic equation

    Quote Originally Posted by anthonye View Post
    ok got it thank you both.
    I want to make sure you understand that x=2/3 is NOT a root.

    $$\left(x+\frac{2}{3}\right)^2 \left|_{x=\frac{2}{3}}\right.=\frac{16}{9} \neq 0$$
    Last edited by romsek; February 19th 2014 at 01:19 PM.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,795
    Thanks
    1574

    Re: quadratic equation

    Quote Originally Posted by HallsofIvy View Post
    You start by learning what a "perfect square" is! For any numbers, a and b, $(a+ b)^2=a^2+ 2ab+ b^2 $. The quadratic here is $9x^2+ 12x+ 4 $. Clearly, $a^2= x^2 $ and $b^2= 4$ so a= 3x and b= 2. From that 2ab= 2(3x)(2)= 12x. The whole point of "completing the square" is adding and subtracting that "2ab" part. I don't see how you can learn "completing the square" without first knowing what you want to get.
    Actually $\displaystyle a^2 = 9x^2 = (3x)^2 $.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Senior Member
    Joined
    Jul 2010
    Posts
    412
    Thanks
    5

    Re: quadratic equation

    Thank you all again just one last thing to do with the discriminant.
    if its equal to zero or a perfect square the quadratic is factorable is the correct?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 9
    Last Post: June 4th 2012, 03:26 PM
  2. Quadratic Equation
    Posted in the Algebra Forum
    Replies: 6
    Last Post: January 15th 2012, 07:08 PM
  3. [SOLVED] Quadratic Equation
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: April 8th 2011, 08:26 AM
  4. Quadratic Equation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: October 17th 2010, 12:23 AM
  5. Replies: 3
    Last Post: April 25th 2010, 04:53 PM

Search Tags


/mathhelpforum @mathhelpforum