Solving Complex Equations: A Case Study on Quadratic Equations with Square Roots

In this article, we will explore a complex equation that involves both a number and its square root. We will walk through the detailed process of solving it, ensuring we identify the correct solution without any extraneous answers.

Introduction

Many mathematical problems, especially in algebra, involve equations with square roots. These can be particularly challenging to solve, but with a systematic approach, they can be daunting problems that become manageable.

A Sample Problem

Consider the following problem: Twice a number plus the square root of the number is twelve minus the square root of the number. Formally, we need to solve the equation:

[2n sqrt{n} 12 - sqrt{n}]

Step-by-Step Solution

Let's define our unknown number as n. The problem can be expressed as:

2n sqrt{n} 12 - sqrt{n}

First, let's rearrange the equation to isolate terms involving the square root of n:

2n sqrt{n} - sqrt{n} 12

This simplifies to:

2n 12 - 2sqrt{n}

Nex let's isolate sqrt{n} by moving all sqrt{n} terms to one side:

2n 2sqrt{n} 12

This can be further simplified by dividing the entire equation by 2:

n sqrt{n} 6

Next, let's isolate sqrt{n} by moving n to the right side:

sqrt{n} 6 - n

To eliminate the square root, square both sides of the equation:

n (6 - n)^2

Expanding the right side:

n 36 - 12n n^2

Now, rearranging the terms to form a quadratic equation:

n^2 - 13n 36 0

Next, we factor the quadratic equation:

(n - 9)(n - 4) 0

Setting each factor to zero gives us the possible solutions:

n - 9 0 rarr; n 9

n - 4 0 rarr; n 4

Verification of Solutions

To ensure these solutions are valid, we will check both in the original equation:

For n 9:

2(9) sqrt{9} 12 - sqrt{9}

18 3 12 - 3

21 ne; 9

This does not hold true, so n 9 is not a solution.

For n 4:

2(4) sqrt{4} 12 - sqrt{4}

8 2 12 - 2

10 10

This holds true, so n 4 is the correct solution.

Therefore, the solution to the equation is n 4.

Final Answer: The number is boxed{4}.

Conclusion

Through this step-by-step process, we have successfully solved a complex equation involving a number and its square root. This method also highlights the importance of checking for extraneous solutions in equations with square roots to ensure accuracy.