Solving Equations: Finding Two Numbers with Given Sum and Product

Suppose you're given that the sum of two numbers is -5 and their product is 5. The challenge is to find these two numbers. This problem can be approached using a system of linear equations and the quadratic formula. Let's dive into the solution step by step.

Setting Up the Equations

Let the two numbers be x and y. According to the problem, we have the following two equations:

x y -5 xy 5

Our first goal is to express one variable in terms of the other using the first equation.

Expressing y in Terms of x

From the first equation:

y -5 - x

This allows us to replace y in the second equation with -5 - x.

Substituting and Simplifying

Substitute y -5 - x into the second equation:

x(-5 - x) 5

Expand and simplify:

-5x - x^2 5

Rearrange to standard quadratic form:

x^2 5x 5 0

Using the Quadratic Formula

Now, we can use the quadratic formula to solve for x. The quadratic formula is given by:

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

Here, a 1, b 5, and c 5. Plugging in these values:

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

x frac{-5 pm sqrt{25 - 20}}{2}

x frac{-5 pm sqrt{5}}{2}

This gives two possible solutions for x:

x_1 frac{-5 sqrt{5}}{2} x_2 frac{-5 - sqrt{5}}{2}

Finding Corresponding y Values

For x_1:

y_1 -5 - x_1 -5 - frac{-5 sqrt{5}}{2} frac{-10 5 - sqrt{5}}{2} frac{-5 - sqrt{5}}{2}

For x_2:

y_2 -5 - x_2 -5 - frac{-5 - sqrt{5}}{2} frac{-10 5 sqrt{5}}{2} frac{-5 sqrt{5}}{2}

Thus, the two numbers are:

left(frac{-5 sqrt{5}}{2}, frac{-5 - sqrt{5}}{2}right) left(frac{-5 - sqrt{5}}{2}, frac{-5 sqrt{5}}{2}right)

Generalization and Verification

Bear in mind that when any two negative numbers are added, the result is a negative number, and the product of any two negative numbers is positive. Therefore, if the problem were posed as finding two numbers whose sum is -5 and product is 5, the numbers must both be negative.

Alternative Methods

While a systematic approach is ideal, for simpler problems, you can guess and check combinations of numbers. However, this method is not always optimal, especially for more complex equations. It's a good idea to familiarize yourself with the quadratic formula and system of linear equations to handle such problems effectively.

By understanding these concepts, you can approach similar problems with confidence. If you need further clarification or have questions, don't hesitate to explore resources online or consult a mathematics tutor!