Simplifying Radical Expressions: Solving sqrt{7 - 4√3}

When dealing with radical expressions, especially those involving square roots, it's often useful to simplify them into a more manageable form. In this article, we'll guide you through the process of simplifying the expression sqrt{7 - 4√3}, using a step-by-step approach that involves algebraic manipulation and the quadratic formula.

Introduction

The expression sqrt{7 - 4sqrt{3}} can appear complex at first glance, but by expressing it in the form sqrt{a} - sqrt{b}, we can simplify it effectively. This article will explore the mathematical steps to achieve this simplification, making use of the system of equations and the quadratic formula.

Step-by-Step Process

Step 1: Assuming the Form

Let's assume that the given expression can be written as:

sqrt{7 - 4sqrt{3}} sqrt{a} - sqrt{b}

Step 2: Squaring Both Sides

Squaring both sides of the equation gives:

7 - 4sqrt{3} (sqrt{a} - sqrt{b})^2 a b - 2sqrt{ab}

Step 3: Equating Rational and Irrational Parts

By equating the rational and irrational parts, we get two separate equations:

a b 7 -2sqrt{ab} -4sqrt{3} rarr; sqrt{ab} 2sqrt{3}

Step 4: Squaring Both Equations

Squaring the second equation gives:

ab 12

Step 5: Solving the System of Equations

We now have a system of equations to solve:

a b 7 ab 12

These equations can be solved by transforming them into a quadratic equation. Using the quadratic formula:

x (-b ± sqrt{b^2 - 4ac}) / 2a

The quadratic equation formed is:

x^2 - (a b)x ab 0 rarr; x^2 - 7x 12 0

Solving this, we get:

x (7 ± sqrt{49 - 48}) / 2 (7 ± 1) / 2

This yields:

x (7 1) / 2 4 x (7 - 1) / 2 3

Thus, we find that a 4 and b 3 (or vice versa).

Step 6: Final Simplification

Therefore, the expression simplifies to:

sqrt{7 - 4sqrt{3}} sqrt{4} - sqrt{3} 2 - sqrt{3}

Conclusion

The final answer is:

sqrt{7 - 4sqrt{3}} 2 - sqrt{3}

This article has demonstrated a systematic approach to simplifying the given radical expression using algebraic manipulation and the quadratic formula. Such techniques are crucial in simplifying complex mathematical expressions and can be applicable to a wide range of similar problems.