the sum of two distinct numbers is 20,and their G.M is 6. find the numbers
To find the two numbers, we can use the given information that their sum is 20 and their geometric mean (G.M) is 6.
Let's assume the two numbers are x and y. We are given that x + y = 20 and the geometric mean is √(xy) = 6.
To solve this system of equations, we can use substitution or elimination method. Let's use the substitution method.
From the equation x + y = 20, we can rearrange it to express y in terms of x: y = 20 - x.
Substituting this value of y into the equation for the geometric mean, we have:
√(x(20 - x)) = 6.
Squaring both sides of the equation, we get:
x(20 - x) = 36.
Expanding the equation, we have:
20x - x^2 = 36.
Rearranging the equation to a quadratic form, we have:
x^2 - 20x + 36 = 0.
Now we can factor this quadratic equation as:
(x - 18)(x - 2) = 0.
Setting each factor to zero, we get the possible values for x:
x - 18 = 0 or x - 2 = 0.
Solving these equations gives us two solutions:
x = 18 or x = 2.
Plugging these values back into the equation y = 20 - x, we can find the corresponding values for y:
For x = 18, y = 20 - 18 = 2.
For x = 2, y = 20 - 2 = 18.
Therefore, the two numbers are 18 and 2.