Product of two consecutive positive integers is 306. Find them.
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Step-by-Step Solution
Step 1: Define variables
We are looking for two consecutive positive integers. We can represent the first integer as x and the next consecutive integer as x+1.
Step 2: Formulate the equation
The problem states that the product of these two consecutive integers is 306. Therefore, we multiply x by (x+1) and set the product equal to 306.
Step 3: Expand and rearrange the equation
Expand the left side of the equation by distributing x to both terms inside the parenthesis. Then, subtract 306 from both sides to set the quadratic equation to zero, which is the standard form ax2+bx+c=0.
Step 4: Solve the quadratic equation by factorization
We need to find two numbers that multiply to give −306 and add up to 1 (the coefficient of x). These numbers are 18 and −17. We then split the middle term and factor by grouping to find the possible values for x.
Step 5: Find the possible values for x
By the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible values for x.
Step 6: Select the correct value for x
The problem specifies that the integers are positive. Therefore, we choose the positive value for x, which is 17. The value x=−18 is discarded because it is not a positive integer.
Step 7: Find the two consecutive integers
Substitute the chosen value of x=17 back into our definitions for the two integers. The first integer is 17, and the second integer is 17+1=18.
Step 8: Verify the solution
To verify our answer, we multiply the two integers we found. Since their product is 306, our solution is correct.