Solving the Equation (n - 9)2 n - 3

Today, we will delve into solving the equation (n - 9)2 n - 3, a common algebra problem that can be expressed in different ways. The goal is to find the value of n that satisfies this equation. Let's break down the process step-by-step and discuss potential confusion points.

Step-by-Step Solution

The given equation is:

(n - 9)2 n - 3

Let's solve it systematically:

Expand the left side of the equation using the FOIL (First, Outer, Inner, Last) method:

(n - 9)2 (n - 9)(n - 9) n^2 - 18n 81

Subtract (n - 3) from both sides:

n^2 - 18n 81 - (n - 3) 0

This simplifies to:

n^2 - 19n 84 0

Factor the quadratic equation:

(n - 12)(n - 7) 0

Solve for n by setting each factor equal to zero:

If (n - 12) 0, then n 12

If (n - 7) 0, then n 7

Validation of the Solutions

To ensure our solutions are correct, let's validate them by substituting n 12 and n 7 back into the original equation:

If n 12:

(12 - 9)2 32 9

12 - 3 9

Both expressions equal 9, so n 12 is a valid solution.

If n 7:

(7 - 9)2 (-2)2 4

7 - 3 4

Both expressions equal 4, so n 7 is also a valid solution.

Potential Misinterpretations

It's important to be clear about the intended equation. For example, if the equation was meant to be:

(n - 9)2 n - 81 (which is the square of 9, i.e., 81)

(n - 9)2 n - 81

n2 - 18n 81 n - 81

n2 - 19n 162 0 (which has no real solutions)

Or if the equation was meant to be:

(n - 9) √(n - 3) (which is the square root of (n - 3))

This would require a different approach, typically involving isolating the square root and squaring both sides.

Therefore, it's crucial to clarify the exact form of the equation before solving it.

Conclusion

In conclusion, the equation (n - 9)2 n - 3 has two valid solutions: n 12 and n 7. Misinterpretations can lead to different results, underlining the importance of careful problem formulation. If you encounter a similar problem, make sure to double-check the intended equation to ensure accurate solutions.