Answer :
Let x represent the number.
four times the number = 4 * x = 4x
The reciprocal of the number = 1/x
If four times a number is added to the reciprocal of that number the result is 5, it means that
4x + 1/x = 5
Multiplying both sides of the equation by x, we have
4x * x + 1/x * x = 5 * x
4x^2 + 1 = 5x
Subtracting 5x fromboth sides of the equation, we have
4x^2 - 5x + 1 = 5x - 5x
4x^2 - 5x + 1 = 0
We would solve for x by applying the method of factorisation. We would find two terms whose sum or difference is - 5x and whose product is 4x^2. The terms are - 4x and - x. By replacing - 5x with - 4x - x, we have
4x^2 - 4x - x + 1 = 0
By factorising, we have
4x(x - 1) - 1(x - 1) = 0
Since x - 1 is common, we have
(x - 1)(4x - 1) = 0
x - 1 = 0 or 4x - 1 = 0
x = 1 or 4x = 1
x = 1 or x = 1/4
We would check these values.
For x = 1,
4 * 1 + 1/1 = 4 + 1 = 5
This is correct
For x = 1/4,
4 * 1/4 + 1/(1/4) = 1 + 4 = 5
It is also correct
Thus, the numbers are 1 and 1/4
