Solving the Cubic Equation \( x^3 - x^2 = 18 \)

A Comprehensive Guide to Finding All Real and Complex Roots

Key Takeaways

• Utilize the Rational Root Theorem to identify potential real roots of the cubic equation.
• Factor the polynomial once a real root is found to simplify the equation into lower-degree polynomials.
• Solve quadratic factors using the quadratic formula to find complex roots when necessary.

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Introduction to Cubic Equations

Cubic equations, characterized by the highest exponent of three on the variable, present a fascinating yet challenging aspect of algebra. The general form of a cubic equation is: \[ ax^3 + bx^2 + cx + d = 0 \] where \( a \neq 0 \). Solving cubic equations involves finding all roots (real and complex) that satisfy the equation. In this guide, we will methodically solve the cubic equation \( x^3 - x^2 = 18 \) by employing systematic algebraic techniques.

Step 1: Rewriting the Equation in Standard Form

To start, it's essential to express the equation in its standard form, setting it equal to zero: \[ x^3 - x^2 - 18 = 0 \] This standardization simplifies the process of applying root-finding methods and polynomial factoring techniques.

Step 2: Applying the Rational Root Theorem

Understanding the Rational Root Theorem

The Rational Root Theorem is a powerful tool in identifying potential rational roots of a polynomial equation. It states that any possible rational root, expressed as \( \frac{p}{q} \), must satisfy: \[ p \text{ is a factor of the constant term, and } q \text{ is a factor of the leading coefficient.} \] For our equation \( x^3 - x^2 - 18 = 0 \), the constant term (\( d \)) is -18, and the leading coefficient (\( a \)) is 1.

Identifying Potential Rational Roots

Given the constant term and leading coefficient, the possible values for \( p \) and \( q \) are:

• Factors of -18 (\( p \)): \( \pm1, \pm2, \pm3, \pm6, \pm9, \pm18 \)
• Factors of 1 (\( q \)): \( \pm1 \)

This results in potential rational roots: \[ \pm1, \pm2, \pm3, \pm6, \pm9, \pm18 \]

Testing Potential Roots

We proceed by substituting each potential root into the original equation to verify if it satisfies the equation.

Potential Root (\( x \))	Substitution Result	Is it a Root?
1	\( 1^3 - 1^2 - 18 = 1 - 1 - 18 = -18 \)	No
-1	\( (-1)^3 - (-1)^2 - 18 = -1 - 1 - 18 = -20 \)	No
2	\( 2^3 - 2^2 - 18 = 8 - 4 - 18 = -14 \)	No
-2	\( (-2)^3 - (-2)^2 - 18 = -8 - 4 - 18 = -30 \)	No
3	\( 3^3 - 3^2 - 18 = 27 - 9 - 18 = 0 \)	Yes
-3	\( (-3)^3 - (-3)^2 - 18 = -27 - 9 - 18 = -54 \)	No
6	\( 6^3 - 6^2 - 18 = 216 - 36 - 18 = 162 \)	No
-6	\( (-6)^3 - (-6)^2 - 18 = -216 - 36 - 18 = -270 \)	No
9	\( 9^3 - 9^2 - 18 = 729 - 81 - 18 = 630 \)	No
-9	\( (-9)^3 - (-9)^2 - 18 = -729 - 81 - 18 = -828 \)	No
18	\( 18^3 - 18^2 - 18 = 5832 - 324 - 18 = 5490 \)	No
-18	\( (-18)^3 - (-18)^2 - 18 = -5832 - 324 - 18 = -6174 \)	No

From the table, we observe that \( x = 3 \) is the only rational root that satisfies the equation.

Step 3: Factoring the Polynomial

Having identified \( x = 3 \) as a root, we can express the cubic polynomial as a product of \( (x - 3) \) and a quadratic polynomial. This process simplifies the cubic equation, making it easier to find the remaining roots.

Performing Polynomial Division

We can perform polynomial long division or synthetic division to factor out \( (x - 3) \) from the cubic polynomial:

(x^3 - x^2 - 18) ÷ (x - 3) = x^2 + 2x + 6
  

Thus, the equation becomes: \[ (x - 3)(x^2 + 2x + 6) = 0 \]

Step 4: Solving the Quadratic Factor

The remaining quadratic equation \( x^2 + 2x + 6 = 0 \) can be solved using the quadratic formula:

\[ x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} \]

Here, \( a = 1 \), \( b = 2 \), and \( c = 6 \).

Calculating the Discriminant

The discriminant (\( \Delta \)) is: \[ \Delta = b^2 - 4ac = 2^2 - 4(1)(6) = 4 - 24 = -20 \] Since the discriminant is negative, the quadratic equation has two complex conjugate roots.

Finding the Complex Roots

Substituting the values into the quadratic formula: \[ x = \frac{ -2 \pm \sqrt{-20} }{2(1)} = \frac{ -2 \pm 2i\sqrt{5} }{2} = -1 \pm i\sqrt{5} \]

Therefore, the complex roots are: \[ x = -1 + i\sqrt{5}, \quad x = -1 - i\sqrt{5} \]

Summary of All Roots

The cubic equation \( x^3 - x^2 = 18 \) has the following roots:

• Real Root: \( x = 3 \)
• Complex Roots: \( x = -1 + i\sqrt{5} \) and \( x = -1 - i\sqrt{5} \)

Graphical Representation

Visualizing the cubic equation helps in understanding the nature of its roots. The graph of \( y = x^3 - x^2 - 18 \) will intersect the x-axis at \( x = 3 \), indicating the real root. The absence of additional x-axis intersections implies that the other two roots are complex.

Verification of the Roots

To ensure the accuracy of the derived roots, we substitute them back into the original equation:

Verifying the Real Root (\( x = 3 \))

\[ 3^3 - 3^2 = 27 - 9 = 18 \] Since the equation holds true, \( x = 3 \) is indeed a valid root.

Verifying the Complex Roots

Substituting \( x = -1 + i\sqrt{5} \) and \( x = -1 - i\sqrt{5} \) into the original equation will satisfy it due to their derivation from the quadratic factor. However, verifying complex roots involves more advanced calculations in the realm of complex numbers, which reaffirm their validity.

Alternative Methods for Solving Cubic Equations

While the Rational Root Theorem and factoring provide a straightforward approach to solving cubic equations, several other methods can be employed, especially when rational roots are not evident.

1. Graphical Method

Plotting the cubic function can offer insights into the number of real roots and their approximate locations. Tools like graphing calculators or software can assist in this visualization.

2. Cardano's Method

Cardano's formula provides an analytical solution to cubic equations. This method involves converting the cubic equation into a depressed cubic and then solving it using radicals. However, it's more complex and often unnecessary when simpler methods like factoring are applicable.

3. Numerical Methods

Techniques such as Newton-Raphson or the bisection method can approximate roots with high precision. These methods are particularly useful for equations where analytical solutions are difficult to obtain.

Applications of Cubic Equations

Cubic equations have widespread applications across various fields:

• Physics: Modeling phenomena like projectile motion and oscillations.
• Engineering: Designing structures and analyzing forces.
• Economics: Optimizing profit functions and market analyses.
• Computer Graphics: Rendering curves and surfaces.

Conclusion

Solving the cubic equation \( x^3 - x^2 = 18 \) showcases the elegance of algebraic methods in unraveling complex mathematical problems. By systematically applying the Rational Root Theorem, factoring, and the quadratic formula, we identified one real root and two complex roots. Understanding these foundational techniques not only aids in solving similar equations but also lays the groundwork for more advanced mathematical concepts.

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