Solving n^3 - 2 36 Using Factorization Method: A Comprehensive Guide

Introduction

Solving cubic equations such as n^3 - 2 36 can be broken down into manageable steps using the factorization method. This article will guide you through the process, explaining multiple approaches including factorization and squaring techniques, to help you understand and solve similar equations effectively.

Squaring Both Sides

Consider the given equation:

[i] n^3 - 2 36 To solve, first add 2 to both sides to isolate the n^3 term:

[i] n^3 38 By taking the cube root of both sides, we can directly solve for n without needing to factorize:

[i] n 38^(1/3)

Since 38^(1/3) is not a whole number, the solution is typically left in this form or evaluated using a calculator.

Factorization Method

For solving equations in the form n^3 - k l, the factorization method, particularly the difference of cubes, can be applied. Here's how to proceed with n^3 - 2 36:

Step 1: Move 36 to the left side of the equation:

[i] n^3 - 38 0

Step 2: Recognize that this is a difference of cubes. Factor it using the identity a^3 - b^3 (a - b)(a^2 ab b^2):

[i] (n - 6)(n^2 6n 36) 0

Step 3: Set each factor equal to zero:

[i] n - 6 0 and n^2 6n 36 0

The first equation is straightforward:

[i] n 6

The second equation, however, represents a quadratic equation. Despite its complex form, it has no real roots, as its discriminant is negative (36^2 - 4 * 1 * 36 ).

Alternative Expansion Method

Another approach is to expand and solve the equation using the FOIL (First, Outer, Inner, Last) method:

Step 1: Rewrite the given equation:

[i] (n - 3)(n - 3) 36

Step 2: Use the FOIL method to expand the left side:

[i] n^2 - 6n 9 36

Step 3: Move 36 to the left side and set the equation to zero:

[i] n^2 - 6n - 27 0

Step 4: Factor the quadratic equation:

[i] (n - 9)(n 3) 0

Step 5: Set each factor equal to zero:

[i] n - 9 0 and n 3 0

Thus, the solutions are:

[i] n 9 or n -3

Proof of Solutions

To verify the solutions:

If n 9:

[i] (9 - 3)^3 36 [i] 6^3 36

Result: 36 36

If n -3:

[i] (-3 - 3)^3 36 [i] -6^3 36

Result: -36 36

Note that these examples highlight a potential oversight in the expansion and factoring process. The correct solution is typically the one that satisfies the original equation, which in this case is n 3.

Conclusion:

Understanding and applying the factorization method and other algebraic techniques can significantly aid in solving cubic equations. By practicing these methods, you can enhance your problem-solving skills and approach equations with confidence.