Problem 426 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{a}+1=\sqrt{a+5}\)

Step-by-Step Solution

Answer

a = 4

Step 1: Isolate the Radical Term

Subtract 1 from both sides of the equation to isolate the radical term on the right side: \(\rightarrow \sqrt{a} = \sqrt{a+5} - 1\)

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square roots: \(\rightarrow (\sqrt{a})^2 = (\sqrt{a+5} - 1)^2\)

Step 3: Simplify Both Sides

Simplify the squared terms: \(\rightarrow a = (a+5) - 2\sqrt{a+5} + 1\)

Step 4: Combine Like Terms

Combine like terms to isolate the square root term: \(\rightarrow a = a + 6 - 2\sqrt{a+5} \rightarrow 0 = 6 - 2\sqrt{a+5} \rightarrow 2\sqrt{a+5} = 6\)

Step 5: Solve for the Radical

Isolate \sqrt{a+5} by dividing both sides by 2: \(\rightarrow \sqrt{a+5} = 3\)

Step 6: Square Both Sides Again

Square both sides again to solve for \(a\): \(\rightarrow (\sqrt{a+5})^2 = 3^2 \rightarrow a + 5 = 9\)

Step 7: Solve for \(a\)

Subtract 5 from both sides to solve for \(a\): \(\rightarrow a = 4\)

Key Concepts

isolating the radical term

The first step in solving a radical equation is to isolate the radical term. To do this, we need to get the radical expression by itself on one side of the equation. Starting with the equation \(\sqrt{a}+1=\sqrt{a+5}\), we can subtract 1 from both sides, giving us the isolated radical term: \(\sqrt{a}=\sqrt{a+5}-1\). This simplification makes further steps more manageable and allows us to focus on eliminating the radical.

squaring both sides

After isolating the radical term, the next step is to square both sides of the equation to eliminate the square root. For our equation \(\sqrt{a}=\sqrt{a+5}-1\), we square both sides: \((\sqrt{a})^2 = (\sqrt{a+5} - 1)^2\). This action changes the equation into one without radical symbols: \(a = a + 5 - 2\sqrt{a+5} + 1\). By squaring both sides, we've transformed the equation into a polynomial, which is easier to solve.

simplifying equations

Once we've squared both sides, we need to simplify the resulting equation. The equation \(a = a + 5 - 2\sqrt{a+5} + 1\) can be simplified by combining the constants: \(a = a + 6 - 2\sqrt{a+5}\). At this stage, you might notice that we can subtract \(a\) from both sides: \(0 = 6 - 2\sqrt{a+5}\). Now, we further simplify by isolating the term with the radical: \(2\sqrt{a+5} = 6\).

combining like terms

Combining like terms in our simplified equation involves grouping constant terms together and keeping variable terms separate. For the equation \(2\sqrt{a+5} = 6\), we need to solve for \(\sqrt{a+5}\). Dividing both sides by 2, we get: \(\sqrt{a+5} = 3\). This step ensures that we have nicely grouped constants and the variable term isolated, making the equation easier to solve in the next operations.

solving for variables

Finally, to solve for the variable \(a\), we square both sides again. From the equation \((\sqrt{a+5})^2 = 3^2\), we get: \(a+5 = 9\). Subtracting 5 from both sides, we find \(a = 4\). This final solution tells us the value of the variable \(a\). Always remember to check your solution by substituting \(a = 4\) back into the original equation to ensure both sides are equal, validating the solution.