Question 298742
Find a quadratic equation that has the solution set (4/3, 2).
<pre><font size = 4 color = "indigo"><b>
     x = {{{4/3}}}        x = 2
    3x = 4         x = 2
3x - 4 = 0     x - 2 = 0
    (3x - 4)(x - 2) = 0
        3x<sup>2</sup> - 10x + 8 = 0 

Please notice that this is simply the exact reverse of 
the step by step procedure for solving the resulting 
equation, which would be

        3x<sup>2</sup> - 10x + 8 = 0 
    (3x - 4)(x - 2) = 0
3x - 4 = 0     x - 2 = 0
    3x = 4         x = 2
     x = {{{4/3}}}        x = 2
</pre></font></b>
And...
A rectangular picture is 8 by 10 inches. The frame has the same width all the way around. The frame and the picture together have a total area of 168 square inches. What is the width of the frame.
<pre><font size = 4 color = "indigo"><b>
here is the picture and frame:

{{{drawing(355+5/9,400,-2,14,-2,16,
red(locate(6,2,red(8)), locate(10,7,red(10)),
red(rectangle(2,2,10,12)), red(line(4,3,6,5)), red(line(6,5,8,3)), 
red(line(6,5,6,9), circle(6,10,1)), line(4,7,8,7)), green(rectangle(0,0,12,14))   )}}}

Let the width of the frame be x
Draw in the 4 square corners, each of which has
side of length x, the width of the frame:

{{{drawing(355+5/9,400,-2,14,-2,16,
red(locate(6,2,red(8)), locate(10,7,red(10)),
red(rectangle(2,2,10,12)), red(line(4,3,6,5)), red(line(6,5,8,3)), 
red(line(6,5,6,9), circle(6,10,1)), line(4,7,8,7)), green(rectangle(0,0,12,14)),
green(rectangle(0,0,2,2)),green(rectangle(10,0,12,2)),green(rectangle(10,12,12,14)),green(rectangle(0,12,2,14)),
green(locate(1,0,x), locate(6,0,8), locate(11,0,x), locate(12.3,1.5,x),
locate(12,7,10), locate(12.3,13.5,x))

   )}}}

The total bottom width of the framed picture is gotten by adding
the green dimensions across the bottom 

<font color = "green">x + 8 + x</font> or 2x + 8

The total height of the framed picture is gotten by adding
the green dimensions along the right edge:

<font color = "green">x + 10 + x</font> or 2x + 10

Using the formula

Area = (bottom width) times (vertical height) we have the equation:

168 = (2x + 8)(2x + 10)

Do you know how to simplify and solve that equation?

If not post again asking how to solve it.  The

solutions you get are x = 2 and x = -11.

We ignore the negative solution and conclude that
the frame is 2 inches wide.

Edwin</pre></font></b>