Solving X² - 10 = 30x A Step-by-Step Guide

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In the realm of algebra, solving quadratic equations is a fundamental skill. A quadratic equation, characterized by its highest power of the variable being 2, can take various forms, and finding its solution set involves determining the values of the variable that satisfy the equation. This article delves into the process of solving the quadratic equation x² - 10 = 30x, providing a step-by-step guide and explaining the underlying concepts.

Understanding Quadratic Equations

Before we embark on solving the equation at hand, let's first grasp the essence of quadratic equations. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and 'x' is the variable we seek to find. The solutions to a quadratic equation are also known as its roots or zeros, representing the points where the equation's graph intersects the x-axis.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations, each with its own strengths and suitability depending on the equation's specific form. The most common methods include:

  1. Factoring: This method involves expressing the quadratic equation as a product of two linear factors. Setting each factor to zero then yields the solutions.

  2. Completing the Square: This technique involves manipulating the equation to form a perfect square trinomial on one side, allowing for easy extraction of the roots.

  3. Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of its factorability. It is expressed as:

    x = (-b ± √(b² - 4ac)) / 2a

  4. Graphical Method: This approach involves plotting the quadratic equation's graph and identifying the points where it intersects the x-axis, representing the solutions.

Solving x² - 10 = 30x: A Step-by-Step Approach

Now, let's tackle the equation x² - 10 = 30x. Our goal is to find the values of 'x' that make this equation true. We'll employ the quadratic formula method, as it provides a reliable solution for any quadratic equation.

1. Rearranging the Equation

The first step is to rearrange the equation into the standard quadratic form ax² + bx + c = 0. To do this, we subtract 30x from both sides of the equation:

x² - 30x - 10 = 0

Now, we can identify the coefficients: a = 1, b = -30, and c = -10.

2. Applying the Quadratic Formula

The quadratic formula is our key to unlocking the solutions. Plugging in the coefficients we identified, we get:

x = (-(-30) ± √((-30)² - 4 * 1 * -10)) / (2 * 1)

Simplifying this expression, we have:

x = (30 ± √(900 + 40)) / 2

x = (30 ± √940) / 2

3. Simplifying the Radical

The radical √940 can be simplified by factoring out the largest perfect square. In this case, 940 = 4 * 235, so:

√940 = √(4 * 235) = 2√235

Substituting this back into our expression for x, we get:

x = (30 ± 2√235) / 2

4. Obtaining the Solutions

Finally, we can simplify the expression by dividing both terms in the numerator by 2:

x = 15 ± √235

This gives us two distinct solutions:

x₁ = 15 + √235

x₂ = 15 - √235

Therefore, the solution set of the equation x² - 10 = 30x is {15 - √235, 15 + √235}.

Verifying the Solutions

To ensure the accuracy of our solutions, we can substitute them back into the original equation and verify that they satisfy it. Let's check both solutions:

For x₁ = 15 + √235:

(15 + √235)² - 10 = 30(15 + √235)

225 + 30√235 + 235 - 10 = 450 + 30√235

450 + 30√235 = 450 + 30√235 (Verified)

For x₂ = 15 - √235:

(15 - √235)² - 10 = 30(15 - √235)

225 - 30√235 + 235 - 10 = 450 - 30√235

450 - 30√235 = 450 - 30√235 (Verified)

Both solutions satisfy the original equation, confirming their correctness.

Conclusion

Solving quadratic equations is a crucial skill in algebra, with various methods available to tackle different forms of equations. In this article, we successfully determined the solution set of the equation x² - 10 = 30x using the quadratic formula, arriving at the solutions 15 + √235 and 15 - √235. By understanding the underlying principles and applying the appropriate techniques, you can confidently navigate the world of quadratic equations and unlock their solutions.

Quadratic equations are a cornerstone of algebra, and mastering their solution is crucial for success in higher-level mathematics. This detailed exploration of the equation x² - 10 = 30x provides a comprehensive guide, ensuring a solid understanding of the quadratic formula and its application. Remember, the quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of its complexity. Understanding the steps involved, from rearranging the equation to simplifying the radical, will empower you to confidently tackle quadratic equations in various contexts.

The solution set of the equation x² - 10 = 30x is {15 - √235, 15 + √235}. This means that these two values of x are the only values that will make the equation true. We found these solutions by using the quadratic formula, a fundamental tool for solving quadratic equations. The quadratic formula allows us to find the solutions even when the equation cannot be easily factored. In this case, the quadratic formula provided us with the precise values of x that satisfy the equation. Understanding and applying the quadratic formula is an essential skill for anyone studying algebra and beyond.

This exploration highlights the importance of the quadratic formula in finding the solution set for quadratic equations. By rearranging the equation into the standard form and applying the quadratic formula, we were able to determine the two solutions: 15 + √235 and 15 - √235. These solutions represent the points where the graph of the quadratic equation intersects the x-axis. The quadratic formula provides a systematic approach to solving quadratic equations, ensuring that we can always find the solutions, even when they are not immediately obvious. This method is a cornerstone of algebraic problem-solving.

By mastering the quadratic formula and understanding its application, you can confidently solve a wide range of quadratic equations. The ability to manipulate equations, identify coefficients, and apply the quadratic formula is a fundamental skill in mathematics. This detailed solution of x² - 10 = 30x serves as a valuable example, demonstrating the power and versatility of the quadratic formula. Remember to always check your solutions by substituting them back into the original equation, ensuring accuracy and a thorough understanding of the problem-solving process.