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The product of two consecutive positive integers is 277. If the smaller of these two integers is represented by x, which of the options below will correspond to the equation for finding out the value of x?
x2+x+277=0
x2тИТx+277=0
x2тИТxтИТ277=0
x2+xтИТ277=0
x2+xтИТ277=0
Since the integers are consecutive, write the product as x2+x. Set this equal to the product given (277) and bring 277 to the left side to equate the quadratic expression to zero.
Two consecutive positive integers are considered. The smaller integer is x. Their product is 277.
x(x+1)=277
Since the integers are consecutive, write the product as x2+x. Set this equal to the product given (277) and bring 277 to the left side to equate the quadratic expression to zero.
Students often mistakenly write x(xтИТ1)=277 by assuming x is the larger integer, or they fail to subtract 277 correctly when moving it to the left side of the equation.
Define the integers
If the smaller integer is x, then the next consecutive integer is (x+1).
Integer1тАЛ=x,Integer2тАЛ=x+1
Formulate the product equation
The problem states the product of these two integers is 277. Multiplying them gives the quadratic expression.
x(x+1)=277
Expand and rearrange
Distributing the x yields x2+x=277. Subtracting 277 from both sides forms the standard quadratic equation ax2+bx+c=0.
x2+xтИТ277=0
D is correct because expanding the product x(x+1)=277 results in x2+xтИТ277=0.
This logic is foundational for solving word problems involving areas or ages where quadratic equations represent the relationship between variables.
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