# Product Sum Calculator + Online Solver With Free Steps

A Product Sum Calculator is used to find the two unknown numbers when their product and sum are provided. The calculator is useful when the sum and product of any two variables or numbers are known and the integers which have produced the sum and product are to be found.

Performing mathematical functions is hard but solving them in reverse order is even harder and a tiresome task. The process involves a lot of arithmetic operations that make solving such questions a tedious task for you.

The Product Sum Calculator makes such kinds of tasks easier as you simply need to enter the problems and the solution is provided by the calculator just in seconds. The calculator provides a direct answer if the functions are entered into the calculator correctly.

This calculator provides the solution by simply adding the numbers or functions into the input boxes. Once the entries are submitted the output window appears with the results.

## What Is the Product Sum Calculator?

A Product Sum Calculator is a useful online calculator that comes in handy for determining which two integers were operated to produce the entered sum and product.

It is useful to operate any kind of product or addition function be it in the numerical or algebraic form. The Product Sum Calculator works in your browser and uses the internet to perform the given mathematical problems efficiently. These problems can be solved by hand which proves to be very lengthy and time-consuming.

The Product Sum Calculator has been designed to find the original numbers let them be x and y. The product and sum of these two unknown numbers are used to find the values by performing basic substitution techniques. The obtained answers can be used to verify the solution by inputting them into original equations.

The calculator is useful in solving not only simple numerical problems but also those containing variables and exponents. The Product Sum Calculator is designed to ease the task of performing the reverse of multiplication and addition.

You can input both the functions into the calculator in the boxes labeled as product and Sum. On submitting, an output tab opens with the answer in the form of values assigned to separate variables x and y.

## How To Use the Product Sum Calculator?

You can use the Product Sum calculator by first finding the product and the sum of unknown variables and then entering the product and sum into the specified fields on the calculator’s screen. The output screen shows those values of the unknown variables. A Product Sum Calculator is very easy to use and efficient in its working.

The following steps must be performed to use the online Product / Sum Calculator:

### Step 1

Consider the product and a sum that is the result of multiplication and addition of the same two values.

### Step 2

Enter the product in the box given in front of the title Product. It can be a perfect square or a simple multiple of two integers.

### Step 3

Enter the sum in the box titled Sum. The sum can be of two integers or two algebraic expressions.

### Step 4

Press Submit to view the result. On clicking the button, a new result window will pop up on your screens showing the desired results.

### Step 5

The output window appears in a separate tab with the required results. The two unknown values are found by the calculator and are expressed as integers. Both of them are assigned to two different variables like x and y.

### Step 6

Other Product Sum problems can also be solved in the same fashion by using this calculator.

It should be considered that the Product Sum Calculator can be used to find out the solutions of simple numerical products and sums as well as those containing variables and algebraic expressions.

## How Does the Product Sum Calculator Work?

A Product Sum Calculator works by performing the arithmetic function of product and sum in reverse. While performing this task manually, many algebraic and other mathematical operations need to be performed in a backward manner such as reverse multiplication or addition. The following two methods are applied:

### Finding Numbers Given Their Product and Sum

If a product and sum are known, the two values that were multiplied or added respectively to produce these results can be calculated. The equations will need to be solved by adding, subtracting, multiplying, dividing, and substituting numbers of the product into the sum or vice versa.

### The Solution to Product Sum of Quadratic Equations

Quadratic equations can be solved either by solving the equations through the addition/subtraction method or by using the substitution or elimination method.

Polynomial and trinomial equations can be solved by breaking down the middle term by the factorization method. For the equation:

$a x^2+b x+c$

The middle term of the equations is the product of the coefficients $a$ and $c$. The sum of the two integers that are obtained by decomposing the middle term, when added gives the middle term b as a result.

## Why a Product Sum Calculator Is Needed

A Product Sum Calculator is needed because of its ability to simplify the complex task of finding the values that produce a certain product and sum. For instance, while solving a problem like this :

If the sum of two numbers is 65 and their product is 156. Find out the two numbers.

Solving it manually requires following steps:

Let the two integers be x and y. Hence,

x+y = 65

$xy = 156$     or  $x= \dfrac{156}{y}$

Putting the value of $x$ in equation x + y = 65.

$\dfrac{156}{y} + y = 65$

157y = 65

y = 0.414013

Putting value of y in equation xy = 156.

x * 0.414013 = 156

$x = \dfrac{156}{0.414013}$

x = 376.7998

However, by using the Product Sum Calculator, all these lengthy steps can vanish and just by clicking one button, you can have your solution.

The product sum technique is used to find the actual numbers that have undergone the multiplication or addition operations. This helps in cross-checking the solution as well as determining the unknown numbers when their product and sum are known.

## Solved Examples

Here are some of the examples of finding the numbers when their product and sum are given. These examples have been solved using the calculator and show how the Product Sum Calculator works.

### Example 1

Find two numbers whose sum is 12 and the product is 36.

### Solution

#### Step 1

Enter 36 into the box titled Product.

#### Step 2

Enter 12 into the box titled Sum.

#### Step 3

Press Submit so that the result appears on the output screen.

#### Result

The result appearing on the output screen is:

x = 6

y = 6

Hence, when x and y both are equal to 6, the product and sum come out to be 36 and 12 respectively.

### Example 2

If the product of two values is $a^2 – b^2$ and their sum is 2a. What are the two values?

### Solution

Enter both the product and sum in the Product Sum Calculator. The output window shows the following results:

#### Result

The two values will be:

x = a – b

y = a + b

or

x = a + b

y = a – b

The answers given above are the values that can produce the product of $a^2 – b^2$  and the sum 2a.

### Example 3

Consider the following:

Product:

x * y = 55

Sum:

x + y = 16

Find the values that produce the product and the sum given above.

### Solution

When you input the values given in the question into the Product Sum Calculator, the following solution is displayed in the output window:

#### Result

The answer can be written in two ways. These are:

The values of x and y can be:

x = 5

y = 11

The pair can also be:

x = 11

y = 5

This is the exact form of the solution.

The approximate form of the answer can also be observed in the output window. If one exists for the given solution, you can see the option on the screen for finding the approximate value. There is another option named More Digit. If the solution can be expressed in a more accurate form, then by selecting the More digits option, more digits after the decimal point can be seen and a more accurate value can be achieved.

The detailed solution for this example is given as:

x* y = 55

x + y = 16

$x = \dfrac{ 55 }{ y }$

Putting the value of x in the equation of sum to find the value of y:

$\dfrac{55}{ y} + y = 16$

$y^2 + 55 = 16y$

$y^2 – 16y + 55 = 0$

Now, breaking the middle term to find the solution for y:

$y^2 -11y -5y + 55 = 0$

y(y – 11) – 5( y – 11) = 0

The values of y are given as:

y = 11

y = 5

Substituting the values of y in $x = \dfrac{55}{y}$ to find the value of x.

The values of  x are given as:

x= 5

x = 11

So, the values of the unknown variables x and y are either x=5, y=11 or x=11 and y=5.

### Example 4

The product of two numbers is $a^4-b^4$ and their sum is $2a^2$. What are the values that are multiplied and added respectively to produce these values as the answer?

### Solution

In the space given for Product enter $a^4-b^4$ and in the space for Sum input $2a^2$. The following result appears on the output screen.

#### Result

The answer is expressed in the following two ways. One way is to express the answer as:

$x = a^2 – b^2$

and

$y = a^2 + b^2$

The other way can be:

$x = a^2 + b^2$

and

$y = a^2 – b^2$

So, the two values that multiply together to make $a^4-b^4$ and add to form $2a^2$ are  $x = a^2 – b^2 \; and \; y = a^2 + b^2$ or $x = a^2 + b^2 \; and \; y = a^2 – b^2$.

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