Problem 88 - Algebra A Combined Function

Question

$$ \text { Solve: } \sqrt{\sqrt{x+3}+\sqrt{x}}=\sqrt{3} $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Simplify the Equation

Begin by squaring both sides of the equation to remove the outer square root. The equation is \( \sqrt{\sqrt{x+3}+\sqrt{x}}=\sqrt{3} \). Squaring both sides gives: \( \sqrt{x+3} + \sqrt{x} = 3 \).

Step 2: Isolate One Square Root

Take one of the square root terms, say \( \sqrt{x+3} \), and isolate it. We do this by expressing \( \sqrt{x+3} \) in terms of \( \sqrt{x} \), leading to \( \sqrt{x+3} = 3 - \sqrt{x} \).

Step 3: Square Again to Eliminate the Remaining Square Root

Square both sides again to eliminate the square root on the left. This gives us: \( x + 3 = (3 - \sqrt{x})^2 \). Now, expand the right-hand side: \( x + 3 = 9 - 6\sqrt{x} + x \).

Step 4: Simplify and Solve for \( x \)

After simplification, the term \( x \) cancels out on both sides: \( 3 = 9 - 6\sqrt{x} \). Rearranging gives: \( 6\sqrt{x} = 6 \). Divide both sides by 6 to find \( \sqrt{x} = 1 \).

Step 5: Solve for \( x \)

Square both sides to solve for \( x \), resulting in \( x = 1^2 = 1 \).

Step 6: Verify the Solution

Substitute back \( x = 1 \) into the original equation: \( \sqrt{\sqrt{1+3} + \sqrt{1}} = \sqrt{\sqrt{4} + 1} = \sqrt{2 + 1} = \sqrt{3} \). Since both sides are equal, \( x = 1 \) is the correct solution.

Key Concepts

Radical Equations

Solving radical equations involves the manipulation of equations that include square roots or other mathematical roots. The goal is to isolate the variable. For instance, solving an equation like \( \sqrt{\sqrt{x+3}+\sqrt{x}}=\sqrt{3} \) requires carefully removing each radical.
If more than one radical exists, like two square roots here, isolate one radical to prevent errors. This is achieved by squaring each side to minimize complexity.
Use squaring multiple times, as seen in this problem, but ensure you assess the result following each squaring. This process repeats until all variables are isolated and radicals removed.

• Step 1 focuses on removing the outer square root by squaring to simplify the operation and equation.
• In Step 3, square both sides again to eliminate another radical, further simplifying to find the true value of \( x \).

Be cautious; squaring both sides can introduce extraneous solutions. Always verify any answers by substituting them back into the original setting to confirm accuracy.

Square Roots

Square roots often indicate needing the original number that, when multiplied by itself, results in the given value. It can introduce complexity into equations.
In our problem, look at \( \sqrt{x+3} + \sqrt{x} = 3 \). Both expressions contain square roots, adding layers of difficulty. To clear these layers, square the sides to eliminate these root signs.
This method allows a straightforward calculation when managed well.

• Use properties of square roots wisely; remember \( \sqrt{a} + \sqrt{b} eq \sqrt{a+b} \).
• Once isolated, square roots reveal hidden values. This property is paramount here, allowing us to disentangle terms.

Attention to detail within expansion processes prevents traditional pitfalls, like oversight of \( (\sqrt{a + b})^2 \) versus \( (a + b) \). Be vigilant about these differences as skipping careful checks leads to incorrect conclusions.

Step-by-Step Solutions

Step-by-step solutions are vital for revealing clear and logical methods for solving complex equations. These solutions help break down a problem into digestible stages.
For the equation \( \sqrt{\sqrt{x+3}+\sqrt{x}}=\sqrt{3} \), each step demystifies the manipulation required. Start by tackling the outermost layer of the equation and move inward.
This strategy helps contain complexity at each phase, maintaining equation integrity and goal focus.

• Step 2 isolates one square root, reducing the equation's complexity and making further simplifications more manageable.
• Step 4 then combines and simplifies terms to a manageable equation. This careful management enables the discovery of real solutions.

Executing each step with precision paves an ordered pathway, ensuring an efficient and successful resolve. Always verify results to remove any erroneous solutions and confirm the final solution is correct.