Question

the difference between two numbers is 9 and the product and the product of the number is 162 find the two numbers

Answer 1

9, 18

Explanation:

lets say the smaller number is x, then the larger number is x+9

we know x * (x+9) = 162.

expand to get x^2 + 9x - 162 = 0, factor or use the quadratic formula to get x^2 + 9x -162 = (x + 18) (x - 9) = 0, so x = 9 or -18.

I'm assuming the two numbers must be positive, so x = 9, and the larger number is 9+9 = 18.

Answer 2

the two numbers are x = -81 - 81*sqrt(7) and y = (-1 - sqrt(7)) / 2.

Explanation:

To find the two numbers, we can use the formula for the sum of two numbers:

x + y = z

Where x and y are the two numbers and z is their sum.

In this case, we know that the difference between the two numbers is 9, so we can set x to be the larger of the two numbers and y to be the smaller number:

x - y = 9

We can also use the fact that the product of the two numbers is 162 to set up an equation:

xy = 162

Now we can solve for x and y by substituting the second equation into the first and solving for x:

x = 162/y

Substituting this expression for x into the first equation gives us the following:

(162/y) - y = 9

We can simplify this equation to:

162 - y^2 = 9

We can solve for y by completing the square:

y^2 - y + 4.5 = 0

We can use the quadratic formula to find the solutions for y:

y = (1 +/- sqrt(1 - 414.5)) / (2*1)

This simplifies to:

y = (-1 +/- sqrt(7)) / 2

Since y is the smaller number, we need to choose the negative solution:

y = (-1 - sqrt(7)) / 2

We can use this value of y to find the value of x:

x = 162/y = 162/(-1 - sqrt(7))/2

This simplifies to:

x = (-81 - 81*sqrt(7))/2

So the two numbers are x = -81 - 81*sqrt(7) and y = (-1 - sqrt(7)) / 2.