# Word Problems

## Word Problems

In word problem exercises, we should translate the text into a quadratic equation with one unknown.
The key to solving these types of problems lies in trying to represent both variables with a single unknown $X$.
After representing the two variables with one unknown, we will continue with the quadratic equation, which is created based on the verbal text of the problem.
We will find the solutions and go back to check what exactly the question was.

## Let's look at an example

The sum of both numbers is $3$
The sum of both numbers raised to the second power is $6.5$.

Solution:

How do we start?
First, we will represent both numbers with a single variable:
The first number: $X$
The second number: $3-X$

How do we describe the second number?

Let's say we had called it $Y$. We could have said, according to the problem's data: $x+y=3$, now let's solve for $Y$ we will get to the representation $Y=3-X$
Therefore, the second number is $3-X$
Next, we will make a quadratic equation based on the rest of the data from this problem.
We will square each of the numbers and add them, their sum must be $6.5$ according to what is stated in the problem.
$X^2+(3-x)^2=6.5$
$x^2+9-6x+x^2=6.5$
We will combine like terms, transpose members and obtain:
$2x^2-6x+2.5=0$
We will solve the equation based on the quadratic formula or the trinomial and it will give us:
$X=2.5 ,0.5$

Observe: These are not the two numbers we are looking for!
These are $2$ options for the first number represented with the $X$.
Now let's see what the two numbers are when the first one is $2.5$ then we will see what the two numbers are when the first one is $0.5$.
When the first number is $2.5$ and the second is $3-X$
The second number is: $0.5$,$3-2.5=0.5$
When the first number is $0.5$ and the second is $3-X$
The second number is: $2.5$,$3-0.5=2.5$

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