Problem 97 - Algebra 2

Question

Solve. Check for extraneous solutions. $$ \sqrt{x^{2}+9}=x+1 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\). After checking, it is confirmed that this is not an extraneous solution.

Step 1: Isolate the Square Root

Start by isolating the square root on one side of the equation. This gives the equation \(\sqrt{x^2 + 9} = x + 1\).

Step 2: Remove the Square Root

To remove the square root, square both sides of the equation. This yields \(x^2 + 9 = (x + 1)^2\). We can simplify the right hand side to get \(x^2 + 9 = x^2 + 2x + 1\).

Step 3: Simplify the Equation

Next, note that there are \(x^2\) terms on both sides of the equation, hence you can subtract \(x^2\) from both sides to simplify into \(9 = 2x + 1\).

Step 4: Solve for x

To solve for x, we subtract 1 from both sides to get \(8 = 2x\), and then divide both sides by 2 to get \(x = 4\).

Step 5: Check for Extraneous Solutions

Now check this solution in the original equation. Plugging it back into the original equation yields \(\sqrt{4^2 + 9} = 4 + 1\), which simplifies to \(\sqrt{25} = 5\). Both sides are equal, so \(x = 4\) is a valid solution.

Key Concepts

Extraneous Solutions

When solving radical equations, it is crucial to check for extraneous solutions. An extraneous solution is one that appears valid throughout the mathematical manipulation steps but ultimately does not satisfy the original equation.

This situation often occurs when squaring both sides of an equation, as this can introduce solutions that were not part of the initial problem. Hence, after finding potential solutions, you must always plug them back into the original equation to verify their validity.

• If both sides of the original equation remain equal with the proposed solution, it means the solution is valid.
• If not, it means those solutions are extraneous and should be disregarded.

Isolating the Square Root

The starting point in solving radical equations typically involves isolating the square root expression. This step is necessary because it sets the stage for removing the square root itself.

To do this, ensure the square root is the only term on one side of the equation. In the example provided, the expression \( \sqrt{x^2 + 9} = x + 1 \) already has the square root isolated. This step simplifies subsequent operations by clearing the way for squaring both sides without complicating additional terms from interfering.

• Examine the equation for any terms outside of the square root that need to be moved to the other side.
• In some equations, you might need to perform addition or subtraction to achieve the isolation.

Squaring Both Sides

After isolating the square root, the next crucial step is squaring both sides of the equation. This method effectively eliminates the square root by applying exponent rules.

When squaring the side of the equation containing the square root, the square and the square root operations cancel each other out. For the equation \( \sqrt{x^2 + 9} = x + 1 \), squaring both sides results in \( x^2 + 9 = (x + 1)^2 \).

Be vigilant about correctly expanding terms when squaring expressions. The right side, \( (x + 1)^2 \), becomes \( x^2 + 2x + 1 \). This accuracy is vital to avoid errors in subsequent steps.

• Ensure the squaring is applied symmetrically to both sides of the equation.
• Double-check expansions to maintain equation integrity.

Simplifying Equations

Once the equation is free from square roots, simplifying it can lead directly to solving for the variable. Simplifying involves combining like terms and isolating the variable of interest.

For our exemplar equation, the simplification following squaring both sides was \( x^2 + 9 = x^2 + 2x + 1 \). Observing similar terms, subtract \( x^2 \) from both sides to get \( 9 = 2x + 1 \). These steps reduce the complexity of solving it further for \( x \).

• Always look for terms that can cancel each other out on opposite sides.
• Perform standard operations to isolate the target variable.

Finally, solve for \( x \) by performing basic arithmetic operations, such as subtraction and division, yielding \( x = 4 \). Always remember to verify it back in the original equation to confirm extraneous checks.