Factoring Quadratic Equations: A Step-by-Step Guide with Practical Examples

Factoring is a fundamental skill in algebra, widely used in solving equations and simplifying expressions. In this guide, we will walk through the process of factoring a quadratic equation, specifically the equation 3c2 - 4c 1. Understanding this step-by-step will enhance your overall grasp of algebraic manipulation techniques.

Introduction to Factoring Quadratic Equations

A quadratic equation is any equation that can be written in the form:

ax2 bx c 0

where a, b, and c are constants, and a ≠ 0. Factoring a quadratic equation involves breaking it down into a product of two simpler expressions, known as factors, which, when multiplied together, give the original equation.

Factoring the Quadratic Equation 3c2 - 4c 1

Let's start with the given quadratic equation:

3c2 - 4c 1

Our goal is to find two expressions that multiply together to give us the original equation.

Step 1: Arrange the Equation in Standard Form

Make sure the equation is in the standard form ax2 bx c. In this case, the equation is already in standard form.

Step 2: Find the Factors

For the quadratic equation 3c2 - 4c 1, we need to find two binomials that multiply to give the original equation. Let's assume the two binomials are (ac - m)(bc - n) such that:

(ac - m)(bc - n) 3c2 - 4c 1

We know that:

The product of the leading coefficients (a * b) must be 3 (the coefficient of c2). The product of the constant terms (-m)(-n) must be 1 (the constant term). The sum of the cross products (am bn) must be -4 (the coefficient of c).

By inspection or trial and error, we find that:

(3c - 1)(c - 1)

Multiplying these binomials:

(3c - 1)(c - 1) 3c2 - 3c - c 1 3c2 - 4c 1

This confirms that the factors of the equation 3c2 - 4c 1 are 3c - 1 and c - 1.

Step 3: Verify the Solution

To ensure accuracy, multiply the factors (3c - 1) and (c - 1) to verify that the result matches the original equation:

(3c - 1)(c - 1) 3c2 - 3c - c 1 3c2 - 4c 1

This confirms that the factors (3c - 1) and (c - 1) are correct.

Conclusion

Factoring quadratic equations is a powerful tool in algebra. By breaking down a complex equation into simpler components, we can more easily solve for the unknowns and analyze the equation's properties. The example of 3c2 - 4c 1 demonstrates the importance of understanding the factors of a quadratic equation. Mastering this skill will greatly enhance your ability to solve a wide range of algebraic problems.

Related Keywords

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