Problem 37 - Intermediate Algebra

Question

Solve. \(\sqrt{x+4}=\sqrt{2 x-5}\)

Step-by-Step Solution

Answer

The solution is x=9.

Step 1: Isolate Square Roots

Both sides of the equation already have a square root expression isolated: \[ \sqrt{x+4} = \sqrt{2x-5} \] This means we can proceed to the next step without any additional manipulation.

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square roots.\[ (\sqrt{x+4})^2 = (\sqrt{2x-5})^2 \]This simplifies to:\[ x+4 = 2x-5\]

Step 3: Solve for x

Rearrange the equation to isolate the variable x. Subtract x from both sides:\[ 4 = x - 5 \]Now, add 5 to both sides:\[ x = 9 \]

Step 4: Verify the Solution

Substitute \(x = 9\) back into the original equation to ensure it satisfies both sides.\[ \sqrt{9+4} = \sqrt{2(9)-5} \]Calculate each side separately:Left side: \( \sqrt{13} \) Right side: \( \sqrt{13} \)Both sides are equal, confirming that \( x = 9 \) is a valid solution.

Key Concepts

Square Root Isolation

To solve an equation involving square roots, the first step is to isolate the square root terms on each side of the equation. This means rearranging the equation, if needed, so that the square root is the only expression on each side. By doing this, you set a solid foundation to eliminate the square roots in the following steps.
In our case:

• Both sides already feature a square root expression.
• The equation is given as \( \sqrt{x+4} = \sqrt{2x-5} \), so the square roots are already isolated.

This allows us to move directly to the next phase of solving, which is squaring both sides of the equation.

Squaring Both Sides

Once the square roots are isolated, the next step is to square both sides of the equation. This action serves to eliminate the square roots, turning the equation into one that is easier to handle.

• When you square \( \sqrt{x+4} \), it simplifies to \( x+4 \).
• Similarly, \( \sqrt{2x-5} \) becomes \( 2x-5 \).

Thus, the equation \( \sqrt{x+4} = \sqrt{2x-5} \) transforms into \( x+4 = 2x-5 \). This simplification leads us towards isolating the variable.

Variable Isolation

Having removed the square roots, the equation now reads \( x+4 = 2x-5 \). The task is to isolate the variable \( x \) to find its value.

• Start by subtracting \( x \) from both sides, simplifying the equation to \( 4 = x - 5 \).
• Next, add 5 to both sides to fully isolate \( x \). This results in \( x = 9 \).

By carefully rearranging and solving the equation, we isolate \( x \), determining that \( x = 9 \). The solution appears correct, but confirming through verification is essential.

Solution Verification

Verifying the solution ensures accuracy and confirms that we've solved the equation correctly. This step involves substituting the calculated value of \( x \) back into the original equation and checking if both sides equal.

• Substitute \( x = 9 \) back into the original equation: \( \sqrt{9+4} = \sqrt{2(9)-5} \).
• Calculate both sides of the equation separately: \( \sqrt{13} \) on the left and \( \sqrt{13} \) on the right.

Since both sides are equal, the solution \( x = 9 \) is verified and confirmed as correct, ensuring we've solved the equation accurately and completely.