Problem 70 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt{2 x+1}-\sqrt{x}=1$$

Step-by-Step Solution

Answer

The short answer for the given equation \(\sqrt{2x+1} - \sqrt{x} = 1\) is: \(x = 0, \quad x = 12\).

Step 1: Isolate one of the square roots

First, we'll isolate \(\sqrt{x}\) on one side of the equation: \[\sqrt{2x+1} - \sqrt{x} = 1 \Rightarrow \sqrt{x} = \sqrt{2x+1} - 1\]

Step 2: Square both sides

Now, we'll square both sides of the equation to eliminate the square roots: \[(\sqrt{x})^2 = (\sqrt{2x+1} - 1)^2\] This simplifies to: \[x = (2x+1) - 2\sqrt{2x+1} + 1\]

Step 3: Simplify and isolate the remaining square root

Now, we will simplify the equation and isolate the term with the square root: \[x = 2x - 2\sqrt{2x+1} + 2 \Rightarrow -x = -2\sqrt{2x+1} + 2 \Rightarrow x-2 = 2\sqrt{2x+1}\]

Step 4: Square both sides again

Square both sides of the equation once more to eliminate the square root: \[(x-2)^2 = (2\sqrt{2x+1})^2\] This simplifies to: \[x^2 - 4x + 4 = 8x+4\]

Step 5: Rearrange to a quadratic equation and solve

We'll now rearrange the equation to form a quadratic equation: \[x^2 - 12x = 0\] Now, factor out the common term of \(x\): \[x(x-12) = 0\] So, the possible solutions are: \[x = 0, \quad x = 12\]

Step 6: Check the solutions

Now, we need to check whether the obtained solutions are valid in the original equation. For \(x=0\): \[\sqrt{2 (0) +1} - \sqrt{0} = 1 \Rightarrow \sqrt{1} - 0 = 1 \Rightarrow 1 = 1\] Thus, \(x=0\) is a valid solution. For \(x=12\): \[\sqrt{2 (12) +1} - \sqrt{12} = 1 \Rightarrow \sqrt{25} - \sqrt{12} = 1 \Rightarrow 5 - \sqrt{12} = 1\] Thus, \(x=12\) is also a valid solution.

Step 7: Conclusion

There are two valid solutions to the given equation: \[x = 0, \quad x = 12\]

Key Concepts

Isolating Square Roots

Solving equations that involve square roots can be tricky. The first crucial step is to isolate one of the square roots. This means bringing the square root term to one side of the equation, separate from other terms.
This is important because it allows us to simplify the equation by removing the square root through squaring.

For instance, in the given equation \( \sqrt{2x+1} - \sqrt{x} = 1 \), we isolate \( \sqrt{x} \) by moving it to the right-hand side. This results in:

• \( \sqrt{x} = \sqrt{2x+1} - 1 \)

By isolating \( \sqrt{x} \), we've created a situation where both sides of the equation can be squared in the next step. Remember, with radical equations, isolating is key as it clears the path for further simplifying the problem.

Quadratic Equations

After isolating the square root, the next step often involves squaring both sides of the equation. By doing this, we eliminate the square roots and end up with a quadratic equation.

This process is illustrated in the equation from our problem:

• \( (\sqrt{x})^2 = (\sqrt{2x+1} - 1)^2 \)

This simplifies to a more classic algebraic form, allowing for further manipulation. Eventually, after additional squaring and rearranging, it becomes the quadratic equation \( x^2 - 12x = 0 \).

Once in a quadratic form, solve it either by factoring or using the quadratic formula. In our example, it's convenient to factor:

• \( x(x-12) = 0 \)
• Therefore, the solutions are \( x = 0 \) and \( x = 12 \)

Quadratic equations are essential for finding possible values of \( x \) where the original equation holds true.

Checking Solutions

Checking solutions is an essential step when solving equations with radicals. Why? Squaring both sides of an equation can sometimes introduce extraneous solutions—solutions that fit the squared equation but not the original equation.
To verify each potential solution, substitute them back into the original equation.

Here's how it's done with the solutions \( x = 0 \) and \( x = 12 \):

• For \( x = 0 \), \( \sqrt{2(0) + 1} - \sqrt{0} = 1 \rightarrow 1 - 0 = 1 \), so it works.
• For \( x = 12 \), \( \sqrt{2(12) + 1} - \sqrt{12} = 1 \rightarrow 5 - \sqrt{12} = 1 \), which is also valid.

Both solutions check out, meaning they satisfy the original equation. Always verify to ensure accuracy in your results!