Question 1186641
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The cardboard is 8cm longer than it is wide.  Since the border is uniform, the picture itself is 8cm longer than it is wide.<br>
The area of the cardboard is 40*32=1280 cm^2; the area of the border is 720 cm^2.  So the area of the picture is 1280-720=560 cm^2.<br>
Let x be the width of the picture; then x+8 is the length.  Since the area is 560...<br>
{{{x(x+8)=560}}}
{{{x^2+8x=560}}}
{{{x^2+8x-560=0}}}
{{{(x+28)(x-20)=0}}}<br>
x=-28 or x=20<br>
Obviously the negative answer makes no sense in the problem.  So x=20.<br>
ANSWER:
picture width: x=20cm
picture length: x+8=28cm<br>
Note in solving the problem using formal algebra we had to factor a quadratic expression by finding two numbers whose difference is 8 and whose product is 560.  That's what the original problem required us to do; so the formal algebra didn't get us any closer to the answer.<br>
Trial and error and a bit of mental arithmetic will get us to the solution quickly:
560=56*10 no...
560=14*40 no...
560=28*20 YES!<br>