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

Answer 1

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 2

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

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