Solving for Numbers That Add to -27 and Multiply to -75

Solving for two numbers that add to -27 and multiply to -75 is a classic algebraic problem often encountered in mathematics. Let's break down the solution using both logical reasoning and mathematical techniques, ensuring the content is SEO-friendly and meets Google's standards.

Step-by-Step Solution

The problem requires finding two numbers, let's call them x and y, such that:

x y -27 xy -75

One approach is to express y in terms of x, using the first equation:

y -27 - x

Substitute this into the second equation:

x(-27 - x) -75

Simplifying this, we get:

-27x - x^2 -75

Or rearranging the terms to form a standard quadratic equation:

x^2 27x - 75 0

Using the Quadratic Formula

To solve this quadratic equation, we can apply the quadratic formula, which is given by:

x [-b ± sqrt(b^2 - 4ac)] / (2a)

Here, a 1, b 27, and c -75. Plugging these values into the formula:

x [-27 ± sqrt(27^2 - 4(1)(-75))] / (2(1))

Simplifying under the square root:

x [-27 ± sqrt(729 300)] / 2

x [-27 ± sqrt(1029)] / 2

Simplifying further, we get:

x [-27 ± 32.07802986] / 2

This gives us two solutions:

x ([-27 32.07802986] / 2) 2.53901493 x ([-27 - 32.07802986] / 2) -29.53901493

Therefore, the two numbers are approximately:**2.54 and -29.54**.

Verifying the Solution

To verify, we can check:

2.54 (-29.54) -27 2.54 * (-29.54) -75

Both conditions are satisfied, confirming our solution is correct.

Practical Applications

Understanding how to solve such problems is crucial in various fields, including:

Economics and Finance: Calculating profit and loss, supply and demand analysis. Physics: Solving kinematic equations and dynamic systems. Engineering: Structural analysis and design optimization.

Recognizing these applications can help students and professionals in related fields to better apply their knowledge in real-world scenarios.

Conclusion

Solving for numbers that add to -27 and multiply to -75 involves a series of logical and mathematical steps. By using the quadratic formula, we can find the exact solutions, which are approximately 2.54 and -29.54. Understanding such problems and techniques is invaluable for students and professionals across various disciplines.