SOLUTION: The outside of a picture frame measures 12in by 19in. 78 in^2 of the picture shows. Find the thickness of the frame.

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Question 445637: The outside of a picture frame measures 12in by 19in. 78 in^2 of the picture shows. Find the thickness of the frame.
Answer by chriswen(106) About Me  (Show Source):
You can put this solution on YOUR website!
Let x in be the thickness of the frame.
...
So,
12-2x will be the width of the picture after you take out the frame.
and 19-2x will be the length of the picture.
...
l*w=A
%2819-2x%29%2812-2x%29=78
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Graphical form: %2819-2x%29%2A%2812-2x%29 simplifies to 4%2Ax%5E2-62%2Ax%2B228
  • Text form: (19-2x)*(12-2x) simplifies to 4*x^2-62*x+228
  • Cartoon (animation) form: simplify_cartoon%28+%2819-2x%29%2A%2812-2x%29+%29
    For tutors: simplify_cartoon( (19-2x)*(12-2x) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at %28highlight_red%28+19+%29-2%2Ax%29%2A%2812-2%2Ax%29.
Moved 19 to the right of expression
It becomes %28-2%2Ax%2Bhighlight_green%28+19+%29%29%2A%2812-2%2Ax%29.
Look at %28-2%2Ax%2B19%29%2A%28highlight_red%28+12+%29-2%2Ax%29.
Moved 12 to the right of expression
It becomes %28-2%2Ax%2B19%29%2A%28-2%2Ax%2Bhighlight_green%28+12+%29%29.
Look at highlight_red%28+%28-2%2Ax%2B19%29%2A%28-2%2Ax%2B12%29+%29.
Expanded term %28-2%2Ax%2B12%29 by using associative property on %28-2%2Ax%2B19%29
It becomes .
Look at -highlight_red%28+%28-2%2Ax%2B12%29%2A2%2Ax+%29%2B%28-2%2Ax%2B12%29%2A19.
Expanded term -2%2Ax by using associative property on %28-2%2Ax%2B12%29
It becomes .
Look at highlight_red%28+2+%29%2Ax%2Ahighlight_red%28+2+%29%2Ax-2%2Ax%2A12%2B%28-2%2Ax%2B12%29%2A19.
Multiplied numerator integers
It becomes highlight_green%28+4+%29%2Ax%2Ax-2%2Ax%2A12%2B%28-2%2Ax%2B12%29%2A19.
Look at 4%2Ax%2Ax-highlight_red%28+2+%29%2Ax%2Ahighlight_red%28+12+%29%2B%28-2%2Ax%2B12%29%2A19.
Multiplied numerator integers
It becomes 4%2Ax%2Ax-highlight_green%28+24+%29%2Ax%2B%28-2%2Ax%2B12%29%2A19.
Look at 4%2Ahighlight_red%28+x+%29%2Ahighlight_red%28+x+%29-24%2Ax%2B%28-2%2Ax%2B12%29%2A19.
Reduce similar several occurrences of highlight_red%28+x+%29 to highlight_green%28+x%5E2+%29
It becomes 4%2Ahighlight_green%28+x%5E2+%29-24%2Ax%2B%28-2%2Ax%2B12%29%2A19.
Look at 4%2Ax%5E2-24%2Ax%2Bhighlight_red%28+%28-2%2Ax%2B12%29%2A19+%29.
Expanded term 19 by using associative property on %28-2%2Ax%2B12%29
It becomes 4%2Ax%5E2-24%2Ax-highlight_green%28+19%2A2%2Ax+%29%2Bhighlight_green%28+19%2A12+%29.
Look at 4%2Ax%5E2-24%2Ax-highlight_red%28+19+%29%2Ahighlight_red%28+2+%29%2Ax%2B19%2A12.
Multiplied numerator integers
It becomes 4%2Ax%5E2-24%2Ax-highlight_green%28+38+%29%2Ax%2B19%2A12.
Look at 4%2Ax%5E2-24%2Ax-38%2Ax%2Bhighlight_red%28+19+%29%2Ahighlight_red%28+12+%29.
Multiplied numerator integers
It becomes 4%2Ax%5E2-24%2Ax-38%2Ax%2Bhighlight_green%28+228+%29.
Look at 4%2Ax%5E2-highlight_red%28+24%2Ax+%29-highlight_red%28+38%2Ax+%29%2B228.
Eliminated similar terms highlight_red%28+-24%2Ax+%29,highlight_red%28+-38%2Ax+%29 replacing them with highlight_green%28+%28-24-38%29%2Ax+%29
It becomes 4%2Ax%5E2%2Bhighlight_green%28+%28-24-38%29%2Ax+%29%2B228.
Look at 4%2Ax%5E2%2B%28-highlight_red%28+24+%29-highlight_red%28+38+%29%29%2Ax%2B228.
Added fractions or integers together
It becomes 4%2Ax%5E2%2B%28highlight_green%28+-62+%29%29%2Ax%2B228.
Look at 4%2Ax%5E2%2B%28highlight_red%28+-62+%29%29%2Ax%2B228.
Removed extra sign in front of -62
It becomes 4%2Ax%5E2%2B%28-highlight_green%28+62+%29%29%2Ax%2B228.
Look at 4%2Ax%5E2%2Bhighlight_red%28+%28-highlight_red%28+62+%29%29%2Ax+%29%2B228.
Remove unneeded parentheses around factor highlight_red%28+62+%29
It becomes 4%2Ax%5E2-highlight_green%28+62+%29%2Ax%2B228.
Result: 4%2Ax%5E2-62%2Ax%2B228

Universal Simplifier and Solver


Done!

4%2Ax%5E2-62%2Ax%2B228=78
4%2Ax%5E2-62%2Ax%2B228-78=0
4%2Ax%5E2-62%2Ax%2B150=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-62x%2B150+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-62%29%5E2-4%2A4%2A150=1444.

Discriminant d=1444 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--62%2B-sqrt%28+1444+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-62%29%2Bsqrt%28+1444+%29%29%2F2%5C4+=+12.5
x%5B2%5D+=+%28-%28-62%29-sqrt%28+1444+%29%29%2F2%5C4+=+3

Quadratic expression 4x%5E2%2B-62x%2B150 can be factored:
4x%5E2%2B-62x%2B150+=+4%28x-12.5%29%2A%28x-3%29
Again, the answer is: 12.5, 3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-62%2Ax%2B150+%29

So therefore there are two answers 12.5 and 3.
But, this is dealing with length and area so the answer can't be a negative number. The equation was (19-2x)(12-2x) If the answer was 6 or more then the (12-2x) would make it negative or 0. So therefore, the answer is 3.
The thickness of the frame is 3 in.