How to Find Two Numbers Given Their Sum and Product

Finding two numbers when you know their sum and product is a classic problem in algebra. Let’s break it down step by step so it’s easy to understand.

Step-by-Step Solution

1. Define the Variables

   Let’s denote the two numbers as $x$ and $y$. We’ll use $S$ to represent their sum and $P$ to represent their product. So, we have:

   1. $x + y = S$
   2. $xy = P$

2. Set Up a Quadratic Equation

   To find $x$ and $y$, we can use these two equations to form a quadratic equation. Recall that the sum and product of the roots of a quadratic equation $ax^2 + bx + c = 0$ are given by $-b/a$ and $c/a$, respectively. In our case, the quadratic equation will be:

   $x^2 – Sx + P = 0$

3. Solve the Quadratic Equation

   To solve the quadratic equation, we use the quadratic formula:

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

   For our equation $x^2 – Sx + P = 0$, $a = 1$, $b = -S$, and $c = P$. Plugging these values into the formula, we get:

   $x = frac{S pm sqrt{S^2 – 4P}}{2}$

4. Find Both Numbers

   The solutions to this equation will give us the two numbers $x$ and $y$. So, we have:

   $x = frac{S + sqrt{S^2 – 4P}}{2}$
   $y = frac{S – sqrt{S^2 – 4P}}{2}$

Example Problem

Let’s go through an example to make this clearer. Suppose we know the sum of two numbers is 10, and their product is 21. We need to find these two numbers.

1. Sum ($S$) = 10
2. Product ($P$) = 21

Using the quadratic equation $x^2 – Sx + P = 0$, we get:

$x^2 – 10x + 21 = 0$

Now, we solve this using the quadratic formula:

$x = frac{10 pm sqrt{10^2 – 4 cdot 1 cdot 21}}{2 cdot 1}$
$x = frac{10 pm sqrt{100 – 84}}{2}$
$x = frac{10 pm sqrt{16}}{2}$
$x = frac{10 pm 4}{2}$

So, the solutions are:

$x = frac{10 + 4}{2} = 7$
$y = frac{10 – 4}{2} = 3$

Therefore, the two numbers are 7 and 3.

Special Cases

Case 1: No Real Solution

If $S^2 – 4P < 0$, the numbers cannot be real because the square root of a negative number is not a real number. For example, if $S = 4$ and $P = 5$, the quadratic equation becomes:

$x^2 – 4x + 5 = 0$

Using the quadratic formula:

$x = frac{4 pm sqrt{16 – 20}}{2}$
$x = frac{4 pm sqrt{-4}}{2}$

Since $sqrt{-4}$ is not a real number, there are no real solutions in this case.

Case 2: Perfect Square Discriminant

If $S^2 – 4P$ is a perfect square, the solutions will be rational numbers. For example, if $S = 8$ and $P = 12$, the quadratic equation becomes:

$x^2 – 8x + 12 = 0$

Using the quadratic formula:

$x = frac{8 pm sqrt{64 – 48}}{2}$
$x = frac{8 pm sqrt{16}}{2}$
$x = frac{8 pm 4}{2}$

So, the solutions are:

$x = frac{8 + 4}{2} = 6$
$y = frac{8 – 4}{2} = 2$

Therefore, the two numbers are 6 and 2.

Conclusion

Finding two numbers given their sum and product is a straightforward process once you understand how to set up and solve the corresponding quadratic equation. Remember, the key steps are to define the variables, set up the quadratic equation, and solve it using the quadratic formula. With a bit of practice, you’ll be able to handle this type of problem with ease.

Practice Problems

Try solving these problems on your own:

1. The sum of two numbers is 15, and their product is 56. What are the numbers?
2. The sum of two numbers is 20, and their product is 91. What are the numbers?
3. The sum of two numbers is 12, and their product is 35. What are the numbers?

Good luck, and happy solving!

Citations

1. 1. Khan Academy – Quadratic Equations
2. 2. Math is Fun – Quadratic Equation
3. 3. Purplemath – Solving Quadratic Equations