document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #306948 by chriswen(106) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let x in be the thickness of the frame.
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\n" ); document.write( "So,
\n" ); document.write( "12-2x will be the width of the picture after you take out the frame.
\n" ); document.write( "and 19-2x will be the length of the picture.
\n" ); document.write( "...
\n" ); document.write( "l*w=A
\n" ); document.write( " $\"%2819-2x%29%2812-2x%29=78\"$
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Solved by pluggable solver: EXPLAIN simplification of an expression

Your Result:

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\n" ); document.write( " YOUR ANSWER\n" ); document.write( "

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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.

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DETAILED EXPLANATION

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Look at $\"%28highlight_red%28+19+%29-2%2Ax%29%2A%2812-2%2Ax%29\"$ .
Moved $\"19\"$ to the right of expression
It becomes $\"%28-2%2Ax%2Bhighlight_green%28+19+%29%29%2A%2812-2%2Ax%29\"$ .
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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 $\"%28-2%2Ax%2B19%29%2A%28-2%2Ax%2Bhighlight_green%28+12+%29%29\"$ .
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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

.
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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

.
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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 $\"highlight_green%28+4+%29%2Ax%2Ax-2%2Ax%2A12%2B%28-2%2Ax%2B12%29%2A19\"$ .
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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 $\"4%2Ax%2Ax-highlight_green%28+24+%29%2Ax%2B%28-2%2Ax%2B12%29%2A19\"$ .
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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 $\"4%2Ahighlight_green%28+x%5E2+%29-24%2Ax%2B%28-2%2Ax%2B12%29%2A19\"$ .
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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 $\"4%2Ax%5E2-24%2Ax-highlight_green%28+19%2A2%2Ax+%29%2Bhighlight_green%28+19%2A12+%29\"$ .
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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 $\"4%2Ax%5E2-24%2Ax-highlight_green%28+38+%29%2Ax%2B19%2A12\"$ .
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Look at $\"4%2Ax%5E2-24%2Ax-38%2Ax%2Bhighlight_red%28+19+%29%2Ahighlight_red%28+12+%29\"$ .
Multiplied numerator integers
It becomes $\"4%2Ax%5E2-24%2Ax-38%2Ax%2Bhighlight_green%28+228+%29\"$ .
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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 $\"4%2Ax%5E2%2Bhighlight_green%28+%28-24-38%29%2Ax+%29%2B228\"$ .
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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 $\"4%2Ax%5E2%2B%28highlight_green%28+-62+%29%29%2Ax%2B228\"$ .
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Look at $\"4%2Ax%5E2%2B%28highlight_red%28+-62+%29%29%2Ax%2B228\"$ .
Removed extra sign in front of $\"-62\"$
It becomes $\"4%2Ax%5E2%2B%28-highlight_green%28+62+%29%29%2Ax%2B228\"$ .
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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: $\"4%2Ax%5E2-62%2Ax%2B228\"$
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Universal Simplifier and Solver

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Done!

\n" ); document.write( "\n" ); document.write( " $\"4%2Ax%5E2-62%2Ax%2B228=78\"$
\n" ); document.write( " $\"4%2Ax%5E2-62%2Ax%2B228-78=0\"$
\n" ); document.write( " $\"4%2Ax%5E2-62%2Ax%2B150=0\"$
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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:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
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\n" ); document.write( " For these solutions to exist, the discriminant $\"b%5E2-4ac\"$ should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%28-62%29%5E2-4%2A4%2A150=1444\"$ .
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\n" ); document.write( " Discriminant d=1444 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28--62%2B-sqrt%28+1444+%29%29%2F2%5Ca\"$ .
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\n" ); document.write( " $\"x%5B1%5D+=+%28-%28-62%29%2Bsqrt%28+1444+%29%29%2F2%5C4+=+12.5\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%28-62%29-sqrt%28+1444+%29%29%2F2%5C4+=+3\"$
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\n" ); document.write( " Quadratic expression $\"4x%5E2%2B-62x%2B150\"$ can be factored:
\n" ); document.write( " $\"4x%5E2%2B-62x%2B150+=+4%28x-12.5%29%2A%28x-3%29\"$
\n" ); document.write( " Again, the answer is: 12.5, 3.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-62%2Ax%2B150+%29\"$

\n" ); document.write( "\n" ); document.write( "So therefore there are two answers 12.5 and 3.
\n" ); document.write( "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.
\n" ); document.write( "The thickness of the frame is 3 in. \n" ); document.write( "

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