Solving the Puzzle of Numbers Whose Product is -90 and Sum is -27

Have you ever come across a fascinating math puzzle that intrigued you? In this article, we dive into one such puzzle: discovering two numbers whose product is -90 and whose sum is -27. Let's explore the steps to solve this intriguing puzzle using the power of algebraic equations and a touch of the quadratic formula.

Setting Up the Equations

Let's denote the two numbers as x and y. We know from the problem that:

x * y -90 x y -27

Expressing One Variable in Terms of Another

From the second equation, we can express y in terms of x as follows:

[ y -27 - x ]

Substituting and Simplifying

Now, we substitute this expression for y into the first equation:

[ x(-27 - x) -90 ]

Expanding this equation, we get:

[ -27x - x^2 -90 ]

Rearranging the equation to standard quadratic form:

[ x^2 27x - 90 0 ]

Solving the Quadratic Equation

To solve the quadratic equation x2 27x - 90 0, we use the quadratic formula:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Here, a 1, b 27, and c -90. Let's calculate the discriminant:

[ b^2 - 4ac 27^2 - 4(1)(-90) 729 360 1089 ]

Now, applying the quadratic formula:

[ x frac{-27 pm sqrt{1089}}{2} ]

Since sqrt{1089} 33, we have:

[ x frac{-27 pm 33}{2} ]

Finding the Solutions

This gives us two possible values for x:

x frac{-27 33}{2} 3 x frac{-27 - 33}{2} -30

Now, we can find the corresponding values for y by substituting these values of x back into the equation y -27 - x:

If x 3, then y -27 - 3 -30 If x -30, then y -27 - (-30) 3

Therefore, the two numbers are 3 and -30.

Alternative Approaches

One can also solve this problem using a more intuitive approach by utilizing the factors of 90. The factors of 90 include 1 x 90, 2 x 45, 3 x 30, 5 x 18, 6 x 15, 9 x 10. Given the product is negative and the sum is negative, we need a mix of positive and negative factors. The pair 3 and -30 fits the criteria:

3 x -30 -90 3 (-30) -27

Alternatively, if we use the algebraic method, we can set:

[ x -27 - y ]

Substitute this into xy -90:

[ (-27 - y)y -90 ]

Transforming it into a quadratic equation:

[ y^2 27y - 90 0 ]

Factoring this equation:

[ (y 30)(y - 3) 0 ]

Thus, solving for y:

[ y -30 text{ or } y 3 ]

And substituting back to find x:

When y -30, x -27 - (-30) 3 When y 3, x -27 - 3 -30

The final answer to the puzzle is: 3 and -30.

Conclusion

Through this detailed exploration, we have successfully solved the intriguing puzzle of finding two numbers whose product is -90 and whose sum is -27. This exercise not only hones our algebraic skills but also demonstrates the versatility of problem-solving techniques. Whether you use the quadratic formula or the factorization method, the solution to the puzzle points to the same pair of numbers, 3 and -30. Happy solving!