Problem 33 - Elementary Algebra

Question

For the following problems, solve the square root equations. $$ \sqrt{2 a+9}-\sqrt{a-4}=0 $$

Step-by-Step Solution

Answer

Answer: There is no real solution for this square root equation.

Step 1: Isolate one of the square roots

Add \(\sqrt{a-4}\) to both sides of the equation to isolate the square root term: $$\sqrt{2a+9}= \sqrt{a-4}$$

Step 2: Square both sides of the equation

Now, square both sides of the equation to eliminate the square roots: $$(\sqrt{2a+9})^2 = (\sqrt{a-4})^2$$ This gives us: $$2a+9 = a-4$$

Step 3: Solve for 'a'

Subtract 'a' from both sides to get: $$a + 9 = -4$$ Now, subtract '9' from both sides to solve for 'a': $$a = -4 - 9$$ Which gives us the value for 'a': $$a = -13$$

Step 4: Check the solution for extraneous solutions

Plug the value of 'a' back into the original equation to see if it works: $$\sqrt{2(-13)+9}-\sqrt{(-13)-4}=0$$ Which simplifies to: $$\sqrt{-17}-\sqrt{-17}=0$$ Since the square roots of negative numbers are not real numbers, the solution \(a=-13\) is an extraneous solution. Therefore, there is no real solution for this square root equation.

Key Concepts

Isolating Square Roots in Equations

When you encounter square root equations, one of the important first steps is to isolate the square root terms. This means you need to get the square root by itself on one side of the equation. For example, consider the equation \( \sqrt{2a+9}-\sqrt{a-4}=0 \). Here, the goal is to separate one of the square roots from the rest of the terms. By adding \( \sqrt{a-4} \) to both sides, we isolate one square root: \( \sqrt{2a+9} = \sqrt{a-4} \).
Breaking down the isolation process, consider:

• Identify the terms associated with the square root.
• Use basic arithmetic operations (like addition or subtraction) to move these terms to one side of the equation.

Isolating square roots sets the stage for the next crucial step: eliminating the square root itself. This is achieved through a method called "squaring both sides."

Squaring Both Sides of an Equation

Once a square root is isolated, the next logical step is to eliminate the square root symbol by squaring both sides of the equation. Let's break this down using our previous example:

• We start with \( \sqrt{2a+9} = \sqrt{a-4} \).
• Squaring both sides results in \( (\sqrt{2a+9})^2 = (\sqrt{a-4})^2 \).
• This simplifies to \( 2a+9 = a-4 \).

Squaring is a powerful tool that turns a square root equation into a linear equation, making it much easier to manage. However, be cautious, as squaring can sometimes introduce what's known as "extraneous solutions." It's crucial to verify the solutions afterward because not every result will be valid in the original equation.

Understanding Extraneous Solutions

Extraneous solutions are false results that emerge when solving equations, especially those involving square roots. They appear because squaring both sides of an equation can introduce solutions that do not satisfy the original equation. Using our exercise, when we solved \( 2a+9 = a-4 \), we found \( a = -13 \). However, upon checking:

• Substitute back: \( \sqrt{2(-13)+9} - \sqrt{(-13)-4} = \sqrt{-17} - \sqrt{-17} \)
• This results in square roots of negative numbers, which aren't real numbers.

So, \( a = -13 \) doesn’t work in the original equation. Hence, it's declared an extraneous solution. Always substitute solutions back into the initial equation to ensure their validity, ensuring you only accept those that make the original equation true.