two numbers are reciprocals of each other and whose sum is 3

find the two numbers which the reciprocals of each other and whose sum of 5

first number --- x

2nd number ---- 1/x

x + 1/x = 3
x^2 + 1 = 3x
x^2 - 3x + 1 = 0
using competing the square:

x^2 - 3x + 9/4 = -1 + 9/4
(x + 3/2)^2 = 5/4
x + 3/2 = ± √5/2
x = -3/2 ± √5/2
x = (-3 + √5)/2 or x = (-3 - √5)/2

Oloa

Let's solve this problem step by step.

Let's say the two numbers are x and y, where x and y are reciprocals of each other. This means that if we take the reciprocal of x, we get y, and if we take the reciprocal of y, we get x. Mathematically, this can be expressed as:

1/x = y (equation 1)
1/y = x (equation 2)

We are also given that the sum of these two numbers is 3. Mathematically, this can be expressed as:

x + y = 3 (equation 3)

Now we have a system of equations: equations 1, 2, and 3. We can solve this system of equations to find the values of x and y.

To solve this system, we can use substitution or elimination.

Method 1: Substitution
We can use equation 1 to substitute for y in equation 3:

x + 1/x = 3

To solve this equation, we can multiply through by x to get rid of the fraction:

x^2 + 1 = 3x

Rearrange the equation:

x^2 - 3x + 1 = 0

Now we have a quadratic equation. We can solve this using the quadratic formula or factoring. Once we find the value of x, we can substitute it back into equation 3 to find the corresponding value of y.

Method 2: Elimination
We can eliminate y by multiplying equation 1 by x and equation 2 by y:

x * (1/x) = y * x
y * (1/y) = x * y

Simplifying, we get:

1 = xy (equation 4)
1 = xy (equation 5)

From equations 4 and 5, we can see that xy = 1. Now we can substitute this value into equation 3:

x + 1/x = 3

Solving this equation will give us the values of x and y.

Once we find the values of x and y, we can verify if they are indeed reciprocals of each other by taking their reciprocals and checking if they satisfy the given conditions.