the product of two numbers is 161. twice the smaller is 9 less than the larger

the numbers are x and 2x+9, so

x(2x+9) = 161

Now just solve for x.

Or, note that 161 = 7*23, which fit the conditions above. You'd have had to figure that out to factor the quadratic anyway.

To solve this problem, let's break it down into two equations.

Let's assume the smaller number is 'x' and the larger number is 'y'.

According to the given information, the product of the two numbers is 161. Therefore, the first equation is:
x * y = 161 -- Equation 1

The second piece of information states that twice the smaller number (2x) is 9 less than the larger number (y). This can be written as:
2x = y - 9 -- Equation 2

Now that we have the two equations, we can solve them simultaneously to find the values of 'x' and 'y'.

We can rearrange Equation 2 to express y in terms of x:
y = 2x + 9

Substituting this value of y in Equation 1, we get:
x * (2x + 9) = 161

Expanding the equation:
2x^2 + 9x - 161 = 0

This is a quadratic equation. To solve for its roots, we can use factoring, completing the square, or the quadratic formula.

However, in this case, factoring might not be straightforward. So let's solve the equation using the quadratic formula:

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Applying this to our equation, we have:
a = 2, b = 9, c = -161

x = (-9 ± sqrt(9^2 - 4 * 2 * -161)) / (2 * 2)

Simplifying further:
x = (-9 ± sqrt(81 + 1288)) / 4
x = (-9 ± sqrt(1369)) / 4
x = (-9 ± 37) / 4

This gives us two possible solutions for x:
x = (-9 + 37) / 4 = 28 / 4 = 7
x = (-9 - 37) / 4 = -46 / 4 = -11.5

Now that we have the values of x, we can substitute them back into Equation 2 to find the corresponding values of y.

For x = 7:
y = 2x + 9 = 2 * 7 + 9 = 14 + 9 = 23

For x = -11.5:
y = 2x + 9 = 2 * -11.5 + 9 = -23 + 9 = -14

Therefore, the two numbers that satisfy the given conditions are 7 and 23, or -11.5 and -14.