Solving Equations Using the Quadratic Formula: A Comprehensive Guide

When faced with quadratic equations, the quadratic formula is a powerful tool to find the solutions. This article will explore how to solve equations such as x - 3^2 10 using the quadratic formula and other methods. We'll delve into the steps involved and provide a detailed explanation for each approach.

Introduction to the Quadratic Formula

The quadratic formula is a method used to solve any quadratic equation of the form ax^2 bx c 0. The formula is given by:

x [-b ± sqrt(b^2 - 4ac)] / (2a)

This formula can be applied to simplify and solve complex quadratic equations. Let's see how it is used in practice.

Solving the Equation x - 3^2 10 Using the Quadratic Formula

The given equation is x - 3^2 10. Let's first transform it into a standard quadratic form.

1. We start with x - 3^2 10.

2. Simplify the left side: x - 9 10.

3. Add 9 to both sides to get the quadratic equation: x^2 - 6x - 1 0.

Applying the Quadratic Formula

For the equation x^2 - 6x - 1 0, the coefficients are:

Substitute these values into the quadratic formula:

x [-(-6) ± sqrt((-6)^2 - 4(1)(-1))] / (2 * 1)

Simplify the expression:

x [6 ± sqrt(36 4)] / 2

x [6 ± sqrt(40)] / 2

x [6 ± 2*sqrt(10)] / 2

x 3 ± sqrt(10)

Therefore, the solutions are:

Alternative Methods: Factoring and Square Root Method

While the quadratic formula is effective, there are alternative methods to solve similar equations. Let's explore these methods in detail.

Solving x - 3^2 10 by Factoring

Another way to solve the equation is by factoring. We can rewrite the equation step-by-step:

Factoring the quadratic equation:

Thus, the solutions are:

Solving x - 3^2 10 by Square Root Method

For simplicity, we can also solve the equation using the square root method:

Add 9 to both sides:

Another approach is to consider the square roots directly:

Therefore, the solutions are:

Conclusion

Whether you choose to use the quadratic formula, factoring, or the square root method, solving equations is a fundamental skill in algebra. Each method has its merits, and understanding all of them can greatly enhance your problem-solving abilities. Practice these methods regularly to become more proficient in handling quadratic equations.