Question 11406
When you say 8-inch-square picture, I take it that you mean a picture with the dimensions 8 in by 8 in. If so, the area of that current picture is 64 square inches. I think you're asking for how many inches will have to be added to each side of that square picture so that the area will be enlarged to 100 square inches.


{{{ 8 * 8 = 64 }}} <------- You know that a picture with sides that are 8 inches long has an area of 64 square inches.


{{{ (8 + x)(8 + x) = 100 }}} <----- You wonder how much to add (+x) to the length and width of the picture so that the resulting area will be 100 square inches.


{{{ 64 + 16x + x^2 = 100 }}} <------ FOILed


{{{ x^2 + 16x + 64 = 100 }}} <------ Re-ordered the terms on the left side because it looks better this way.


{{{ x^2 + 16x - 36 = 0 }}} <------ Subtract 100 from both sides. This way, the right side will be zero. We can only solve quadratic equations if the right side is equal to zero!


{{{ (x + 18)(x - 2) = 0 }}} <-------- This is reverse-FOILING, or factoring. What we did was found two integers that when times'd together gives us -36 BUT add up to +16. 18 and -2 do it, so we put it in (x +/- ?)(x +/- ? ) form.


From the above, we can tell that x = -18 or x = 2. Since -18 is NOT a valid value for a measure (because we can't have negative measurements), the 2 is the correct answer. So, if we add 2 to the sides of the 8 x 8 square, we'll end up with a 10 x 10 square, which indeed will have an area of 100 square inches.