The square of one integer is 55 less than the square of the next consecutive integer. Find the lesser integer.

Let x = the lesser integer and let x + 1 = the greater integer. The sentence, "The square of one integer is 55 less than the square of the further consecutive integer," can translate to the equation x² = (x + 1)2 -55. Multiply the binomial (x + 1)² as (x + 1)(x + 1) by using the distributive property:  x² = x² + 1x + 1x + 1 - 55. Combine  such as terms: x² = x² + 2x - 54. Subtract x2 from both sides of the equation: x² - x² = x² - x² + 2x - 54. Add 54 to both sides of the equation: 0 + 54 = 2x - 54 + 54. Divide both sides through 2:  54/2 = 2x/2. The variable is now alone: 27 = x. The lesser integer is 27.