Proofing a quadratic equation
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What is the area of the small rectangle (created by overlapping rectangles) at the center of the sketch?
Please show us what you have tried and exactly where you are stuck.
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pka
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Mentally move the vertical rectangle to one end of the flag. Then that vertical area is \(2x\)
Then the area of the reaming horizontal part is \((3-x)x\)
Now \((3-x)x+2x=~?\)
HallsofIvy
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I see two rectangles, one 3 m by x m, so area 3x square meters, the other 2 m by x m, so area 3x square meters. However, they overlap in the middle so that "x by x" region has been counted twice. We have to subtract that once to so as not to count that overlap twice: 3x+ 2x- x^2= 5x- x^2.
Another way to do this is to think of it as 5 non-overlapping regions. The flag is 3 m long. Subtracting the x m wide middle the remaining 3- x is divided into two rectangles each of length (3-x)/2 and width x so area (3-x)x/2= (3x- x^2)/2. The two rectangles, both left and right, have total area 2(3x- x^2)/2= 3x-x^2. The flag is 2 m in width. Subtracting the x m high middle the remaining 2- x is divided into two rectangles each of length (2- x)/2 and width x so area (2x- x^2)/2. The two rectangles, both above and below, have total area 2(2x- x^2)/2= 2x- x^2. Finally, the middle is "x by x" so has area x^2 meters. Adding all of those the total area of the cross is (3x- x^2)+ (2x- x^2)+ x^2= 5x- x^2.
(And the verb you want is "prove", not "proof".)
Another way to do this is to think of it as 5 non-overlapping regions. The flag is 3 m long. Subtracting the x m wide middle the remaining 3- x is divided into two rectangles each of length (3-x)/2 and width x so area (3-x)x/2= (3x- x^2)/2. The two rectangles, both left and right, have total area 2(3x- x^2)/2= 3x-x^2. The flag is 2 m in width. Subtracting the x m high middle the remaining 2- x is divided into two rectangles each of length (2- x)/2 and width x so area (2x- x^2)/2. The two rectangles, both above and below, have total area 2(2x- x^2)/2= 2x- x^2. Finally, the middle is "x by x" so has area x^2 meters. Adding all of those the total area of the cross is (3x- x^2)+ (2x- x^2)+ x^2= 5x- x^2.
(And the verb you want is "prove", not "proof".)