Question 225245

Solve this quadratic equation using the "multiply to give, add to give" rule:<BR>
3x^2-5x-12=0<BR>
Thanks</I><BR><B>Answer by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(2810)</B></A><IMG SRC=/images/stars/stars-13.gif>&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=/images/aboutme-small.gif BORDER=0 alt="About Me" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=/images/nopicture.jpg> (<A HREF=/cgi-bin/show-question-source.mpl?solution=168124>Show Source</A>): <DIV STYLE="font-family: Comic Sans MS;">
You can <A HREF="/cgi-bin/embed-solution.mpl?show=1&solution=168124">put this solution on YOUR website!</A><BR><IMG SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2-2=0&x=0003 ALIGN=MIDDLE  > <BR>
<PRE><font size = 4 color = "indigo"><b>
Divide every term by 2

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2%2F2-2%2F2=0&x=0003 ALIGN=MIDDLE  >

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-1=0&x=0003 ALIGN=MIDDLE  >

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E2-1%5E2=0&x=0003 ALIGN=MIDDLE  >

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%282x%2B1%29=0&x=0003 ALIGN=MIDDLE  >

2x - 1 = 0      2x + 1 = 0
    2x = 1          2x = -1
     x = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2&x=0003 ALIGN=MIDDLE  >          x =<IMG SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2&x=0003 ALIGN=MIDDLE  > 

-----------------------------------------------------------

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29x%5E2-blue%285%29x-green%2812%29=0&x=0003 ALIGN=MIDDLE  >

Multiply <IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29&x=0003 ALIGN=MIDDLE  > by <IMG SRC=/cgi-bin/plot-formula.mpl?expression=green%28-12%29&x=0003 ALIGN=MIDDLE  >, get <IMG SRC=/cgi-bin/plot-formula.mpl?expression=-36&x=0003 ALIGN=MIDDLE  >

Think of two numbers that multiply to 
give you <IMG SRC=/cgi-bin/plot-formula.mpl?expression=-36&x=0003 ALIGN=MIDDLE  > and add to give you <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28-5%29&x=0003 ALIGN=MIDDLE  >.

Answer: <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28-9%29&x=0003 ALIGN=MIDDLE  > and <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28%22%22%2B4%29&x=0003 ALIGN=MIDDLE  >

So use those numbers <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28-9%29&x=0003 ALIGN=MIDDLE  > and <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28%22%22%2B4%29&x=0003 ALIGN=MIDDLE  >

to write the middle term <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28-5x%29&x=0003 ALIGN=MIDDLE  > as <IMG SRC=/cgi-bin/plot-formula.mpl?expression=blue%28-9x%2B4x%29&x=0003 ALIGN=MIDDLE  >

So

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29x%5E2-blue%285%29x-green%2812%29=0&x=0003 ALIGN=MIDDLE  >

now becomes:

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29x%5E2-blue%289x%2B4x%29-green%2812%29=0&x=0003 ALIGN=MIDDLE  >

Now out of the first two terms on the left 
factor out <IMG SRC=/cgi-bin/plot-formula.mpl?expression=3x&x=0003 ALIGN=MIDDLE  >

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=3x%28x-3%29%2B4x-12=0&x=0003 ALIGN=MIDDLE  >

Out of the last two terms on the left 
factor out <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%22%22%2B4&x=0003 ALIGN=MIDDLE  >

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=3x%28x-3%29%2B4%28x-3%29=0&x=0003 ALIGN=MIDDLE  >

Notice that <IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%28%28x-3%29%29&x=0003 ALIGN=MIDDLE  > occurs in
both terms on the left:

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=3x%2Ared%28%28x-3%29%29%2B4%2Ared%28%28x-3%29%29=0&x=0003 ALIGN=MIDDLE  >

So factor out <IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%28%28x-3%29%29&x=0003 ALIGN=MIDDLE  >

<IMG SRC=/cgi-bin/plot-formula.mpl?expression=red%28%28x-3%29%29%283x%2B4%29=0&x=0003 ALIGN=MIDDLE  >

 x - 3 = 0      3x + 4 = 0
     x = 3          3x = -4
                     x =<IMG SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3&x=0003 ALIGN=MIDDLE  >
Edwin</PRE><BR>