To find two numbers that add up to 2 and multiply to make 8, we can set up the problem with two variables, say x and y.
We know two things:
- x + y = 2
- x * y = 8
From the first equation, we can express y in terms of x:
y = 2 – x
Now, we can substitute this expression for y into the second equation:
x * (2 – x) = 8
Expanding this gives us:
2x – x² = 8
Rearranging it into a standard quadratic equation form:
x² – 2x + 8 = 0
Next, we can use the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / 2a
For our equation, a = 1, b = -2, and c = 8. Plugging these values into the formula gives us:
x = (2 ± √((-2)² – 4 * 1 * 8)) / (2 * 1)
x = (2 ± √(4 – 32)) / 2
x = (2 ± √(-28)) / 2
Since we have a negative number inside the square root, the solutions for x (and therefore y) are not real numbers. This means that there are no two real numbers that satisfy both conditions simultaneously.
In conclusion, there are no two real numbers that add up to 2 and multiply to 8.