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Square roots are mathematical expressions that represent a number which, when multiplied by itself, gives the original number. In the given equation \( \sqrt{k^2 - 4} = 4 - k \), the square root \( \sqrt{k^2 - 4} \) can be understood as the principal (non-negative) root of the expression inside. This is important because it ensures that when squaring both sides of the equation, we focus on balancing an equation rather than just simplifying it by eliminating the root.

It's crucial to remember that squaring both sides of an equation nullifies the square root. This is why the step \( (\sqrt{k^2 - 4})^2 = (4 - k)^2 \) works; squaring a square root simply leaves you with the expression inside. However, one must be cautious since squaring introduces extraneous solutions, which are not necessarily solutions of the original equation. This concept becomes essential when performing the verification step later.

Algebraic manipulation involves rearranging equations to isolate variables. To solve our equation after squaring both sides, you need to be able to expand terms properly and combine like terms.

For the equation \( k^2 - 4 = (4 - k)^2 \), make use of the expansion

• FOIL (First, Outer, Inner, Last method) for \((4 - k)(4 - k)\)
• First, multiply 4 by 4 to get 16
• Outer, multiply 4 by \(-k\) to get \(-4k\)
• Inner, multiply \(-k\) by 4, also \(-4k\)
• Last, multiply \(-k\) by \(-k\) to get \(k^2\)

Simplifying gives us \( k^2 - 4 = 16 - 8k + k^2 \). This may appear lengthy, but with practice, these steps become quicker.

After simplification, combine all like terms on one side to discover that all variable terms \(k^2\) cancel out, leaving a straightforward equation \(8k = 16\). Dividing by 8 simplies this to \(k = 2\). Apart from just solving, ensure during algebraic manipulation that you don't lose information or introduce errors.

Verification is an essential step in solving any mathematical problem to ensure correctness. After determining a possible solution, insert it back into the original equation to confirm it's valid. In this case, substituting \(k = 2\) into \(\sqrt{k^2 - 4} = 4 - k\) doesn’t hold true, since \(\sqrt{4-4} = 0\) is not equal to \(2\).

This check shows that \(k = 2\) is not a valid solution. Failed verification means you must revisit steps or conclude that there is no solution, just as seen here, where extraneous results emerged due to squaring.

• Always use the original equation to verify.
• Prevent mistakes by not skipping this step.

The verification step is helpful in confirming the accuracy and understanding the possibility of no solutions within certain constraints in a given numerical problem.