To find two numbers that multiply to -24 and add up to 5, we can set up a system of equations based on the properties of numbers in a quadratic equation. Specifically, if and are the two numbers, then:
1. (The product of the two numbers)
2. (The sum of the two numbers)
We can solve this using substitution or by using the quadratic formula. Since , we can express in terms of as .
Substituting into the first equation:
Expanding this equation:
Rearranging it to form a standard quadratic equation:
Now, we can solve this quadratic equation either by factoring, if possible, or using the quadratic formula. Let’s check for factorization first:
The factors of -24 that add up to -5 are -8 and 3. Therefore, the equation factors as:
Setting each factor equal to zero gives us the solutions:
Thus, the two numbers are 8 and -3:
• They multiply to
• They add up to
So, the two numbers are 8 and -3.
