Question

which two numbers when multiplied gives 20 but when added gives 9

Answer 1

Let the two numbers be \(x\) and \(y\).

According to the problem, we have the following two equations:

1. \(x \cdot y = 20\)
2. \(x + y = 9\)

We can solve this system of equations. From the second equation, we can express \(y\) in terms of \(x\):

\[
y = 9 - x
\]

Now, we can substitute this expression for \(y\) into the first equation:

\[
x(9 - x) = 20
\]

Expanding this gives us:

\[
9x - x^2 = 20
\]

Rearranging it into standard quadratic form:

\[
x^2 - 9x + 20 = 0
\]

Next, we can factor this quadratic equation. We need two numbers that multiply to 20 and add to -9. The numbers are -4 and -5:

\[
(x - 4)(x - 5) = 0
\]

Setting each factor equal to zero gives us the solutions:

\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

Thus, the two numbers are \(4\) and \(5\).

Checking the conditions:
- Multiplication: \(4 \cdot 5 = 20\)
- Addition: \(4 + 5 = 9\)

Hence, the two numbers are **4 and 5**.