it is important to note that the area of a square is equal to the length of one side raised to the power of 2.
the length of one side of the larger square is unknown-- let's call it x.
the length of one side of the smaller square can be expressed as x-3.
the area of the larger square is x².
the area of the smaller square is (x-3)².
if the sum of the areas is 149, so x²+(x-3)² = 149.
let's expand (x-3)². by "FOILing" that expression, you'll get that (x-3)² = x² -3x -3x +9, which is equal to x²-6x +9.
so x² + x² - 6x + 9 = 149
2x² - 6x + 9 = 149
Subtract each side by 9.
2x² - 6x = 140
factor out a 2. so 2(x² - 3x) = 140
divide both sides by 2. so x² - 3x = 70.
subtract both sides by 70. so x² - 3x - 70 = 0.
this can be written as (x-10)(x+7) = 0.
so x-10 = 0 and x+7 = 0.
so x = 10 and x = -7
we said that x is equal to the length of a side of the larger square, so the length of one side of the larger square can be 10 or -7. but when we're working with measurements, you cannot use a negative number (because you can't have something that is -7 inches long, that doesn't make any sense). so we're going to throw out the -7 and say that x = 10. so the larger square has a side length of 10 inches. we said that the length of a side of the smaller square is x-3, so the length of the smaller square's sides is 7 (which is 10-3).
