The sum of two numerbs is 100. there product is 900, what are hte numbers

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

1. \( x + y = 100 \)
2. \( xy = 900 \)

We can solve for one of the variables using the first equation. Let's solve for \( y \):

\[ y = 100 - x \]

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

\[ x(100 - x) = 900 \]

Expanding this gives:

\[ 100x - x^2 = 900 \]

Rearranging this into a standard quadratic form:

\[ x^2 - 100x + 900 = 0 \]

Now we can use the quadratic formula to solve for \( x \):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this case, \( a = 1 \), \( b = -100 \), and \( c = 900 \):

\[ x = \frac{100 \pm \sqrt{(-100)^2 - 4 \cdot 1 \cdot 900}}{2 \cdot 1} \]

Calculating the discriminant:

\[ x = \frac{100 \pm \sqrt{10000 - 3600}}{2} \]

\[ x = \frac{100 \pm \sqrt{6400}}{2} \]

\[ x = \frac{100 \pm 80}{2} \]

Now, we have two possible solutions for \( x \):

1. \( x = \frac{180}{2} = 90 \)
2. \( x = \frac{20}{2} = 10 \)

Now we can find the corresponding values of \( y \):

• If \( x = 90 \), then \( y = 100 - 90 = 10 \).
• If \( x = 10 \), then \( y = 100 - 10 = 90 \).

Thus, the two numbers are \( 90 \) and \( 10 \).