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Chapter 3: Problem 23

Find all real solutions to each equation. Check your answers. $$\sqrt{n+4}+\sqrt{n-1}=5$$

Short Answer

Expert verified
The real solution is \(n=5\).

Step by step solution

01

- Isolate one of the square roots

Move \(\sqrt{n-1}\) to the other side of the equation: \(\sqrt{n+4} = 5 - \sqrt{n-1}\)
02

- Square both sides

Square both sides to eliminate the square root: \((\sqrt{n+4})^2 = (5 - \sqrt{n-1})^2\). This simplifies to \(n+4 = 25 - 10\sqrt{n-1} + (n-1)\)
03

- Simplify the equation

Combine like terms: \(n + 4 = 24 + n - 10\sqrt{n-1}\). Subtract \(n\) from both sides: \(4 = 24 - 10\sqrt{n-1}\). Subtract 24: \(-20 = -10\sqrt{n-1}\)
04

- Solve for the remaining square root

Divide both sides by -10: \(2 = \sqrt{n-1}\). Then square both sides: \(4 = n - 1\)
05

- Calculate the value of n

Add 1 to both sides: \(n = 5\)
06

- Check the solution

Substitute \(n = 5\) back into the original equation: \(\sqrt{5+4} + \sqrt{5-1} = \sqrt{9} + \sqrt{4} = 3 + 2 = 5\). The equation holds true, so \(n = 5\) is a valid solution.

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Key Concepts

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

Isolating Square Roots
The first step in solving equations involving square roots is to isolate one of the square roots. This means moving terms around so that one square root is by itself on one side of the equation. For the equation \(\sqrt{n+4} + \sqrt{n-1} = 5\), we can move \(\sqrt{n-1}\) to the other side to get \(\sqrt{n+4} = 5 - \sqrt{n-1}\).
This sets us up for the next step, which is to deal with the isolated square root more easily.
Isolating square roots helps simplify the equation. Moving terms around makes it easier to manage the mathematical operations later.
Always strive to have your square root term alone before proceeding further.
Squaring Both Sides
Once we have isolated a square root, the next step is to square both sides of the equation. This action gets rid of the square root. In our example, we had \(\sqrt{n+4} = 5 - \sqrt{n-1}\).
Squaring both sides, we get \((\sqrt{n+4})^2 = (5 - \sqrt{n-1})^2\). This simplifies to \(n+4 = 25 - 10\sqrt{n-1} + (n-1)\).
Squaring both sides will eliminate the square root but be mindful! Sometimes, squaring can introduce extraneous solutions, so always check your answers at the end.
Use Minimized steps and simplifications each time you square both sides to avoid complexities and errors.
Simplifying Equations
Simplification is an essential step after squaring both sides of an equation. This involves combining like terms and reducing the equation to a more manageable form.
For instance, after squaring, we had \(n + 4 = 25 - 10\sqrt{n-1} + (n-1)\). We first combine like terms to get \(n + 4 = 24 + n - 10\sqrt{n-1}\).
Then, we subtract \(n\) from both sides and simplify: \(4 = 24 - 10\sqrt{n-1}\). Further simplification by subtracting 24 from both sides gives \(-20 = -10\sqrt{n-1}\).
Breaking complex equations into simpler forms step-by-step can reduce error chances. Each simplification brings you closer to the solution.
Checking Solutions
After solving the equation, it's crucial to check the solutions. This ensures that no extraneous roots were introduced during the squaring process.
To check the solution \(n = 5\), substitute it back into the original equation: \(\sqrt{5+4} + \sqrt{5-1} = \sqrt{9} + \sqrt{4} = 3 + 2 = 5\).
Since the original equation holds true, \(n = 5\) is a valid solution.
Skipping the verification step can lead to incorrect results, especially for equations involving square roots. Always verify to ensure the accuracy of the derived solution.

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