Solving Cubic and Quadratic Equations: A Comprehensive Guide

Understanding and solving equations is a fundamental skill in mathematics, applicable across various fields from physics to engineering. This guide will help you understand and solve cubic and quadratic equations, using examples to illustrate the process.

Solving a Cubic Equation

Solving a cubic equation involves several steps. Let's start with the equation:

x^3 x^2 - 3x 0

The first step is to move all terms to one side to set the equation to zero:

x^3 x^2 - 3x - 0

Next, factor out the common term:

x(x^2 x - 3) 0

This gives us one answer immediately:

x 0

To solve the remaining quadratic equation x^2 x - 3 0, we can use the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 1, b 1, and c -3. Plugging these values into the formula, we get:

x frac{-1 pm sqrt{1^2 - 4(1)(-3)}}{2(1)}

Simplifying further:

x frac{-1 pm sqrt{1 12}}{2}

x frac{-1 pm sqrt{13}}{2}

The two solutions are:

x frac{-1 sqrt{13}}{2} x frac{-1 - sqrt{13}}{2}

Solving a Quadratic Equation

A quadratic equation is of the form ax^2 bx c 0. We can solve it using the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Let's take the example: x^3 - x x times x^2, which simplifies to:

3 - x x^2

Bringing all terms to one side:

x^2 x - 3 0

Using the quadratic formula with a 1, b 1, and c -3:

x frac{-1 pm sqrt{1 - 4(1)(-3)}}{2}