Question

waht are two numbers their sum is 22 and their product is 120

Answer 1

Let the two numbers be \( x \) and \( y \). According to the problem, we have the following two equations:

1. \( x + y = 22 \)
2. \( x \cdot y = 120 \)

You can solve these equations simultaneously. From the first equation, express \( y \) in terms of \( x \):

\[
y = 22 - x
\]

Now, substitute this expression for \( y \) into the second equation:

\[
x(22 - x) = 120
\]

Expanding this gives:

\[
22x - x^2 = 120
\]

Rearranging the equation, we get a quadratic equation:

\[
x^2 - 22x + 120 = 0
\]

Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -22, c = 120 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-22)^2 - 4 \cdot 1 \cdot 120 = 484 - 480 = 4
\]

Now, applying the quadratic formula:

\[
x = \frac{22 \pm \sqrt{4}}{2 \cdot 1} = \frac{22 \pm 2}{2}
\]

This yields two potential values for \( x \):

1. \( x = \frac{24}{2} = 12 \)
2. \( x = \frac{20}{2} = 10 \)

Now substituting back to find \( y \):

- If \( x = 12 \), then \( y = 22 - 12 = 10 \).
- If \( x = 10 \), then \( y = 22 - 10 = 12 \).

Thus, the two numbers are **10 and 12**.