SOLUTION: Originally the dimensions of a rectangle were 12 cm by 7 cm. When both dimensions were decreased by the same amount, the area of the rectangle decreased by 34 cm^2. Find the dime

Question 243013: Originally the dimensions of a rectangle were 12 cm by 7 cm. When both dimensions were decreased by the same amount, the area of the rectangle decreased by 34 cm^2. Find the dimensions of the new rectangle.
Answer by oberobic(2304) (Show Source): You can put this solution on YOUR website!
Start with the given information:
The original dimensions of a rectangle (which is a well-defined shape with lots of known relationships) are 12 by 7. That means the L=12 and W=7. Area = 12*7 = 84.
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Now for the tricky parts.
If we were to subtract the same unknown amount, let's call it 'x', from both the length and width, the area would be 'reduced by 34'. Substituting we can express this as:
A = (L-x) * (W-x) = 84 - 34 = 50
...
So, now we can solve by substituting what we know about L and W:
(12-x)(7-x)=50
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Multiplying through:
84 - 12x - 7x + x^2 = 50
Rearranging and collecting like terms:
x^2 -19x + 84 = 50
Subtract 50 from both sides:
x^2 - 19x + 34 = 0
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Can we factor 34 into two factors that, when added together, total 19?
Yes. 2 * 17 = 34. 2+17 = 19.
So we factor it, remembering that a negative times a negative is positive, but the sum is negative:
(x - 2)(x - 17) = 0
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That means we have two candidate solutions:
x = 2
x = 17
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But recall we are subtracting 'x' from values 12 and 7 to arrive at a new rectangle.
We cannot have negative sides to a rectangle, so x=2 is the only reasonable answer.
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Of course, we always have to check our work.
Is (12-2)(7-2) = 50?
Yes, it is 10*5 = 50.
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We're not done until we check what the question asks. In this case, it asks what the new dimensions are. So we say "The dimensions of the new rectangle are 10 by 5."
Now we're done.
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