What Two Numbers Multiply to -14 and Add to 8?

Solving mathematical problems can be both challenging and rewarding, especially when it involves finding two numbers that satisfy specific conditions. In this article, we will explore how to find two numbers that multiply to -14 and add to 8. We'll break down the problem into simple steps and explain the mathematical logic behind it.

Understanding the Problem

We are given the task to find two numbers such that the product of these numbers is -14 and their sum is 8. Let's denote the two numbers as x and y. We can set up the following equations based on the given conditions:

x y 8

x * y -14

Approach to the Solution

To solve this problem, we can follow these steps:

Rearrange the first equation to express one variable in terms of another. Here, we can express y in terms of x:

(y 8 - x)

Substitute this expression into the second equation:

((8 - x) * x -14)

Expand and rearrange the equation to standard quadratic form:

(8x - x^2 -14)

(x^2 - 8x - 14 0)

Use the quadratic formula to solve for x:

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

Here, (a 1), (b -8), and (c -14).

Substitute these values into the quadratic formula:

(x frac{8 pm sqrt{(-8)^2 - 4 cdot 1 cdot (-14)}}{2 cdot 1})

(x frac{8 pm sqrt{64 56}}{2})

(x frac{8 pm sqrt{120}}{2})

(x frac{8 pm 2sqrt{30}}{2})

(x 4 pm sqrt{30})

With the values of x found, substitute back into the expression for y:

When (x 4 sqrt{30}), (y 8 - (4 sqrt{30}) 4 - sqrt{30})

When (x 4 - sqrt{30}), (y 8 - (4 - sqrt{30}) 4 sqrt{30})

Therefore, the two numbers are:

(x 4 sqrt{30}), (y 4 - sqrt{30})

or

(x 4 - sqrt{30}), (y 4 sqrt{30})

Decimal Approximations

The decimal approximations of these numbers are:

4 - sqrt{30} -0.597 (to 3 decimal places) 4 sqrt{30} 8.597 (to 3 decimal places)

These numbers satisfy both conditions: their sum is 8 and their product is -14.

Conclusion

Solving mathematical problems, especially involving quadratic equations, can be approached systematically. By setting up and solving the equations step-by-step, we can find the numbers that meet the given conditions.