Question

give me a number that when multiplied together equal 100 but added together equal 21

Answer 1

how about x+y=100 and x*y=21

Answer 2

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

1. \( x \cdot y = 100 \)
2. \( x + y = 21 \)

To solve these equations, we can express \( y \) from the second equation:

\[
y = 21 - x
\]

Next, substitute \( y \) into the first equation:

\[
x(21 - x) = 100
\]

Expanding this gives:

\[
21x - x^2 = 100
\]

Rearranging it into standard form results in:

\[
x^2 - 21x + 100 = 0
\]

Now we can use the quadratic formula to find the values of \( x \):

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

Here, \( a = 1 \), \( b = -21 \), and \( c = 100 \):

\[
x = \frac{21 \pm \sqrt{(-21)^2 - 4 \cdot 1 \cdot 100}}{2 \cdot 1}
\]
\[
x = \frac{21 \pm \sqrt{441 - 400}}{2}
\]
\[
x = \frac{21 \pm \sqrt{41}}{2}
\]

Calculating \( x \):

\[
x_1 = \frac{21 + \sqrt{41}}{2}
\]
\[
x_2 = \frac{21 - \sqrt{41}}{2}
\]

Thus, we have the two possible numbers \( x \) and \( y \):

\[
y = 21 - x
\]
or

\[
y_1 = \frac{21 - \sqrt{41}}{2}, \quad y_2 = \frac{21 + \sqrt{41}}{2}
\]

So the two numbers that multiply to 100 and add to 21 are approximately:

\[
x \approx 19.7, \quad y \approx 1.3
\]
or,
\[
x \approx 1.3, \quad y \approx 19.7
\]

Answer 3

Let's define the two numbers as \( x \) and \( y \). We're given two equations:

1. \( x + y = 100 \)
2. \( x \cdot y = 21 \)

First, we can express \( y \) from the first equation:

\[
y = 100 - x
\]

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

\[
x(100 - x) = 21
\]

Expanding this gives:

\[
100x - x^2 = 21
\]

Rearranging it into standard form results in:

\[
x^2 - 100x + 21 = 0
\]

Now we can use the quadratic formula to find the values of \( x \):

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

Here, \( a = 1 \), \( b = -100 \), and \( c = 21 \):

\[
x = \frac{100 \pm \sqrt{(-100)^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1}
\]
\[
x = \frac{100 \pm \sqrt{10000 - 84}}{2}
\]
\[
x = \frac{100 \pm \sqrt{9916}}{2}
\]

Calculating \( \sqrt{9916} \):

\[
\sqrt{9916} \approx 99.58
\]

Now plugging that back in:

\[
x \approx \frac{100 \pm 99.58}{2}
\]

Calculating both solutions:

1. For the positive root:
\[
x_1 \approx \frac{100 + 99.58}{2} \approx \frac{199.58}{2} \approx 99.79
\]

2. For the negative root:
\[
x_2 \approx \frac{100 - 99.58}{2} \approx \frac{0.42}{2} \approx 0.21
\]

Now we can find \( y \) using \( y = 100 - x \):

1. For \( x_1 \):
\[
y_1 \approx 100 - 99.79 \approx 0.21
\]

2. For \( x_2 \):
\[
y_2 \approx 100 - 0.21 \approx 99.79
\]

Thus, the two numbers that satisfy the equations \( x + y = 100 \) and \( x \cdot y = 21 \) are approximately:

\[
x \approx 99.79, \quad y \approx 0.21
\]
or
\[
x \approx 0.21, \quad y \approx 99.79
\]