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Chapter 2: Problem 36

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt{x+2}=6$$

Short Answer

Expert verified
The real solution of the given radical equation is \(x=34\).

Step by step solution

01

Isolate the square root

To isolate the square root, the equation \(\sqrt{x+2}=6\) is already in the required form. The square root term, \(\sqrt{x+2}\), is on one side of the equation and the constant, \(6\), is on the other side of the equation.
02

Square both sides of the equation

Next, square both sides of the equation to remove the square root. Doing this gives \((\sqrt{x+2})^2 = 6^2\), which simplifies to \(x+2 = 36\).
03

Solving for x

To find the value of \(x\), subtract 2 from both sides of the equation: \(x+2 - 2 = 36 - 2\). This simplifies to \(x=34\).
04

Verify the solution

Finally, it is important to verify the solution by substitifying it back into the original equation. Substituting \(x = 34\) into the equation \(\sqrt{x+2}=6\) gives \(\sqrt{34+2}\), which simplifies to \(\sqrt{36}=6\). Since the left and right side of the equation are now equal, \(x=34\) is indeed a real solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Square Root
Solving radical equations, such as \( \sqrt{x+2}=6 \), begins with isolating the square root on one side of the equation. Isolating the square root is a crucial first step because it sets up the stage for removing the radical, and thus, simplifying the equation.

To isolate the square root, you need to ensure that the square root expression stands alone on one side of the equation. In our example, \( \sqrt{x+2} \) is already by itself on one side, and it鈥檚 equal to \( 6 \) on the other, which means we are ready for the next step. If the square root were not isolated, we'd have to use addition or subtraction to move other terms to the opposite side of the equation.

Remember, if there is a coefficient attached to the square root, you would have to divide both sides by that coefficient to isolate the square root.
Squaring Both Sides
Once the square root is isolated, the next step in solving radical equations is to square both sides. This is done to eliminate the square root, allowing us to solve for the variable normally.

For our example, when we square \( \sqrt{x+2} \), we get \(x+2\). On the other side of the equation, squaring \(6\) gives us \(36\). The equation \( \sqrt{x+2}=6 \) turns into \( x+2 = 36 \) post-squaring. This simplifies to \( x = 34 \) after subtracting \( 2 \) from both sides.

Be mindful when squaring both sides that you square the entire side, not just the terms individually. This is especially important when dealing with more complex equations that have additional terms.
Checking Solutions in Radical Equations
After solving a radical equation, it's essential to check that the solution works in the original equation. This is because squaring both sides can introduce what are known as extraneous solutions 鈥� these are answers that fit the squared equation but not the original radical equation.

To check the solution \( x=34 \) in the equation \( \sqrt{x+2}=6 \) we substitute \(x\) back into the original equation. Replacing \(x \) with \(34\), we get \( \sqrt{34+2} \) which simplifies to \( \sqrt{36} = 6\), confirming that it鈥檚 correct. It鈥檚 always a good idea to check your solution to make sure it does not cause any negative numbers to appear under the radical, which would invalidate the solution in the set of real numbers.

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