Arectangular picture frame measures 10 cm by 12 cm. do you want to triple the frames area by adding the same distance "x" to the length and the width. write and solve an equation to find the value of "x". what are the new dimensions of the picture frame?

If the original frame area is x, then length*width=12*10=120=area=x. By tripling the frame's area, we get 120*3=360. However, we need to add a number x to both the length and width to equal the new area. Since length*width=area, and 10+x is the width and 12+x is the length, we get (10+x)*(12+x)=360. Expanding using the Distributive Property, we get 120+10x+12x+x^2=x^2+22x+120=360. Subtracting 360 from both sides, we get x^2+22x-240=0.Finding 2 numbers that add up to 22 and multiply to -240, we do a bit of guess and check ( we COULD use the quadratic formula too) to find 30 and -8 fit the bill. Therefore, x^2+30x-8x-240=0=
x(x+30)-8(x+30)=(x-8)(x+30). To make the expression equal to 0, since 0 multiplied by anything is 0, we have to figure out when x-8 and x+30 equal 0, which is when x=8 and -30 respectively (To do this, you could add 8 from both sides to turn x-8=0 to x=8, and something similar to -30). Plugging each one in, we get (10+(8))*(12+(8))=18*20=360. To make sure there aren't 2 solutions, we get that the new width is 10-30=-20, which clearly can't be negative , making x equal to 8.