Final answer:
The number that, when added to four times its reciprocal, results in ½ is determined by solving a simple algebraic equation. One solution is found by direct calculation, but we must recognize that a quadratic equation will yield two possible solutions.
Step-by-step explanation:
The student is asking to find a number such that when it is added to four times its reciprocal, the result is ½. This is a typical problem involving algebraic manipulation and reciprocal relationships.
Let the number be denoted as x. The equation representing the problem is:
x + 4(⅖) = ⅗2
To solve for x, we set up the equation:
x + ⅔ = ⅗2
By finding a common denominator, we will have:
(2x + 4) ÷ 2 = ⅗2
Multiplying both sides by 2 to clear the denominator gives us the quadratic equation:
2x + 4 = 17
Subtracting 4 from both sides, we get:
2x = 13
Dividing by 2, we find one value of x as:
x = ⅖4
Since the number has a reciprocal, x cannot be 0. To check for another possible number, we must recognize that a quadratic equation usually has two solutions. We can rewrite our original equation as:
x + ÷ = ⅗2
This shows that we will end up with a quadratic equation upon finding a common denominator and simplifying. The possible numbers are the solutions to the resulting quadratic equation.