Introduction

Finding two numbers that multiply to a specific value and add to another specific value is a common problem in mathematics. This is particularly useful in understanding quadratic equations and solving real-world problems such as optimizing profits or analyzing statistical data.

Solving the Problem

To solve a problem where two numbers x and y multiply to -50 and add to 5, we can follow these steps:

Step 1: Set Up the Equations

Let the two numbers be x and y.

(x cdot y -50) (x y 5)

Step 2: Express One Variable in Terms of the Other

We can express y in terms of x using the second equation:

gy 5 - x

Step 3: Substitute and Simplify

Now substitute this expression for y into the first equation:

(x(5 - x) -50)

Expanding this equation gives:

(5x - x^2 -50)

Rearranging the equation leads to:

(x^2 - 5x - 50 0)

Step 4: Solve the Quadratic Equation

This is a quadratic equation in the form (ax^2 bx c 0), where (a 1), (b -5), and (c -50). We can solve this using the quadratic formula:

(x frac{{-b pm sqrt{b^2 - 4ac}}}{2a})

Substitute the values of (a), (b), and (c):

(x frac{5 pm sqrt{{(-5)^2 - 4 cdot 1 cdot (-50)}}}{2 cdot 1})

Distribute and simplify:

(x frac{5 pm sqrt{25 200}}{2})

(x frac{5 pm sqrt{225}}{2})

(x frac{5 pm 15}{2})

This gives us two solutions:

(x frac{20}{2} 10) (x frac{-10}{2} -5)

Step 5: Find the Corresponding Value of the Other Variable

Substituting back to find y:

If (x 10), then (y 5 - 10 -5) If (x -5), then (y 5 - (-5) 10)

Therefore, the two numbers are 10 and -5.

Verification

To verify, let's check if 10 and -5 satisfy both conditions:

10 (-5) 5 10 cdot (-5) -50

Conclusion

The two numbers that multiply to -50 and add to 5 are 10 and -5. This problem can be solved systematically by setting up equations and using the quadratic formula, making it a valuable tool in algebraic problem-solving.