# Lesson Formula for finding the square of the next integer without a calculator

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This Lesson (Formula for finding the square of the next integer without a calculator) was created by by Jstrasner(106)  : View Source, Show
About Jstrasner: I am a sophomore at Brandeis University, I have been doing this for 3 years now and I teach pre-schoolers and love to help students out.

 If you have a number, such as 16, and you find out the square, which is 256, and you want to find out the square root of 17, which is the next integer up from 16, you would use the following formula: If x+1=y (x being 16, y being 17), then (x^2)+ (x+y) = y^2, or (16^2) + (16+17) = 17^2, 256+33=289, 289=289 Therefore: No matter what numbers you use, as long as they can fit into the formula x+1=y , you can find the square of the higher number, or visa versus, using the equation: (x^2) + (x+y) = y^2 The proof for this formula derives from the rule that y=x+1 (or x=y-1): y = x + 1 => (y^2) = (x+1)^2 => (y^2) = (x^2) + 2x + 1 => (y^2) = (x^2) + x + x +1 => (y^2) = (x^2) + (y-1) + x + 1 => (y^2) = (x^2) + y + x - 1 + 1 => (y^2) = (x^2) + y + x => (x^2) + (x+y) = (y^2)