To find two numbers that multiply to 3 and add to 4, we can set up a system of equations. Let's call the two numbers x and y.
We have:
1. The multiplication equation: x * y = 3
2. The addition equation: x + y = 4
To solve this, we can start with the addition equation. We'll express one number in terms of the other and then substitute it into the multiplication equation. Here's how it works:
From the addition equation:
x + y = 4
y = 4 - x
Now we have y expressed in terms of x. Next, we substitute this expression for y into the multiplication equation:
x * y = 3
x * (4 - x) = 3
Now we have a quadratic equation to solve for x:
x * (4 - x) = 3
4x - x^2 = 3
Rearranging and setting the equation to zero, we get:
x^2 - 4x + 3 = 0
We can factor this quadratic as follows:
(x - 3)(x - 1) = 0
This gives us two solutions:
x = 3 or x = 1
Using the x-values, we can find the corresponding y-values:
If x = 3:
y = 4 - x
y = 4 - 3
y = 1
If x = 1:
y = 4 - x
y = 4 - 1
y = 3
So, the two pairs of numbers that multiply to 3 and add to 4 are (3, 1) and (1, 3). These pairs are interchangeable, but there are no other integer pairs that fulfill both requirements.
