Solving Radical Equations: Finding the Value of x

The problem asks us to find the value of \(x\) in the given radical equation:

\[ \frac{\sqrt{19 - x \sqrt{12}}}{1} = \sqrt 4 - \sqrt 3 \]

Let's simplify the equation step by step to solve for the value of \(x\).

Step 1: Simplify the Right Side of the Equation

The right side of the equation involves simplifying square roots:

\[ \sqrt 4 - \sqrt 3 = 2 - \sqrt 3 \]

So the equation becomes:

\[ \sqrt{19 - x \sqrt{12}} = 2 - \sqrt 3 \]

Step 2: Simplify the Term Under the Square Root on the Left Side

Simplify the \(\sqrt{12}\) term:

\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt 4 \times \sqrt 3 = 2\sqrt 3 \]

Substitute this back into the equation:

\[ \sqrt{19 - x (2\sqrt{3})} = 2 - \sqrt 3 \] \[ \sqrt{19 - 2x\sqrt{3}} = 2 - \sqrt 3 \]

Step 3: Square Both Sides of the Equation

To eliminate the square root on the left side, square both sides of the equation:

\[ (\sqrt{19 - 2x\sqrt{3}})^2 = (2 - \sqrt 3)^2 \]

\[ 19 - 2x\sqrt{3} = (2 - \sqrt 3)^2 \]

Step 4: Expand the Right Side

Expand the square on the right side using the formula \((a-b)^2 = a^2 - 2ab + b^2\):

\[ (2 - \sqrt 3)^2 = 2^2 - 2(2)(\sqrt 3) + (\sqrt 3)^2 \]

\[ (2 - \sqrt 3)^2 = 4 - 4\sqrt 3 + 3 \]

\[ (2 - \sqrt 3)^2 = 7 - 4\sqrt 3 \]

Now substitute this back into the squared equation:

\[ 19 - 2x\sqrt{3} = 7 - 4\sqrt 3 \]

Step 5: Solve for x

We now have a linear equation involving \(x\). Isolate the term with \(x\):

\[ -2x\sqrt{3} = 7 - 4\sqrt 3 - 19 \]

\[ -2x\sqrt{3} = -12 - 4\sqrt 3 \]

Divide both sides by \(-2\sqrt{3}\) to find \(x\):

\[ x = \frac{-12 - 4\sqrt 3}{-2\sqrt 3} \]

We can simplify this expression by dividing both the numerator and the denominator by \(-2\):

\[ x = \frac{\frac{-12}{-2} + \frac{-4\sqrt 3}{-2}}{\frac{-2\sqrt 3}{-2}} \]

\[ x = \frac{6 + 2\sqrt 3}{\sqrt 3} \]

Now, divide each term in the numerator by \(\sqrt 3\):

\[ x = \frac{6}{\sqrt 3} + \frac{2\sqrt 3}{\sqrt 3} \]

\[ x = \frac{6}{\sqrt 3} + 2 \]

To simplify \(\frac{6}{\sqrt 3}\), rationalize the denominator by multiplying the numerator and denominator by \(\sqrt 3\):

\[ \frac{6}{\sqrt 3} \times \frac{\sqrt 3}{\sqrt 3} = \frac{6\sqrt 3}{3} = 2\sqrt 3 \]

Substitute this back into the expression for \(x\):

\[ x = 2\sqrt 3 + 2 \]

Rearranging the terms, we get:

\[ x = 2 + 2\sqrt 3 \]

Conclusion: Value of x

The value of \(x\) that satisfies the given equation is \(2 + 2\sqrt 3\). This matches one of the provided options.

Revision Table: Key Concepts for Radical Equations

Additional Information: Working with Square Roots

When solving equations involving square roots, it's important to remember the properties of radicals:

• The square root of a product is the product of the square roots: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) for non-negative \(a\) and \(b\).
• Adding or subtracting terms with square roots is like combining like terms: you can only combine terms with the same radical part (e.g., \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\), but \(3\sqrt{2} + 5\sqrt{3}\) cannot be simplified further).
• Squaring a term like \( (a - \sqrt{b}) \) or \( (a + \sqrt{b}) \) results in a mix of rational and irrational terms.
• Always check if the value under the square root is non-negative in the original equation's domain. In this case, we squared, so we implicitly assumed \(19 - x\sqrt{12} \ge 0\) and \(2 - \sqrt 3 \ge 0\). Since \(2 = \sqrt{4}\) and \(2 > \sqrt 3\), the right side is positive, which is necessary for it to be equal to a square root.

Understanding how to simplify and manipulate expressions with square roots is fundamental for solving radical equations like this one and finding the value of \(x\).