What Is The Solution Set Of X2 10 30x

What is the Solution Set of x² + 10 = 30x? A Comprehensive Guide

Solving quadratic equations is a fundamental skill in algebra. This article delves into finding the solution set for the quadratic equation x² + 10 = 30x, exploring various methods and providing a detailed, step-by-step explanation. We’ll cover the quadratic formula, factoring, and completing the square, ultimately demonstrating that understanding multiple approaches enhances problem-solving skills.

Understanding the Quadratic Equation

Before diving into the solution, let's understand the structure of the equation: x² + 10 = 30x. This is a quadratic equation because the highest power of the variable 'x' is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

To solve our equation, we first need to rearrange it into the standard form:

x² - 30x + 10 = 0

Now we can clearly identify a = 1, b = -30, and c = 10.

Method 1: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation, regardless of whether it can be easily factored. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting our values (a = 1, b = -30, c = 10) into the formula:

x = [30 ± √((-30)² - 4 * 1 * 10)] / (2 * 1) x = [30 ± √(900 - 40)] / 2 x = [30 ± √860] / 2 x = [30 ± 2√215] / 2 x = 15 ± √215

Therefore, the solution set using the quadratic formula is:

x = 15 + √215 and x = 15 - √215

These are the exact solutions. Approximate decimal values can be obtained using a calculator:

x ≈ 29.99 and x ≈ 0.34

Understanding the Discriminant (b² - 4ac)

The expression inside the square root, b² - 4ac, is called the discriminant. It provides valuable information about the nature of the roots (solutions) of the quadratic equation:

• If b² - 4ac > 0: The equation has two distinct real roots (as in our case).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex roots (involving imaginary numbers).

In our equation, the discriminant is 860 (which is > 0), indicating two distinct real roots.

Method 2: Factoring (if possible)

Factoring involves expressing the quadratic equation as a product of two linear factors. This method is only effective if the quadratic expression can be easily factored. In our case, finding two numbers that add up to -30 and multiply to 10 is not straightforward, making factoring impractical. Therefore, we will proceed with the other methods, since factoring will not yield a simple solution here.

Method 3: Completing the Square

Completing the square is another technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Here's how to complete the square for our equation:

1. Move the constant term to the right side:

   x² - 30x = -10

2. Take half of the coefficient of x (-30), square it ((-15)² = 225), and add it to both sides:

   x² - 30x + 225 = -10 + 225 x² - 30x + 225 = 215

3. Factor the left side as a perfect square:

   (x - 15)² = 215

4. Take the square root of both sides:

   x - 15 = ±√215

5. Solve for x:

   x = 15 ± √215

This yields the same solution set as the quadratic formula:

x = 15 + √215 and x = 15 - √215

Graphical Representation

The solutions to the quadratic equation represent the x-intercepts (where the graph intersects the x-axis) of the parabola y = x² - 30x + 10. Plotting this parabola visually confirms the existence of two distinct x-intercepts, corresponding to the two real roots we calculated. While we won't create a graph here, using graphing software or a calculator can provide a visual representation of the solution set.

Applications of Quadratic Equations

Quadratic equations have widespread applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing bridges, structures, and other architectural elements.
• Economics: Modeling supply and demand, optimizing production costs.
• Computer Graphics: Creating curved shapes and animations.

Understanding how to solve quadratic equations efficiently is crucial in these and other domains.

Choosing the Best Method

The best method for solving a quadratic equation depends on the specific equation.

• Factoring: The quickest method if the equation factors easily.
• Quadratic Formula: Always works, providing exact solutions for any quadratic equation.
• Completing the Square: Useful for specific types of equations and for deriving the quadratic formula itself. It also helps in understanding the geometric interpretation of the equation.

For our equation, x² + 10 = 30x, the quadratic formula or completing the square is the most efficient approach since factoring is not readily apparent.

Conclusion

The solution set for the quadratic equation x² + 10 = 30x is {15 + √215, 15 - √215}. We explored three different methods – the quadratic formula, factoring (although impractical in this specific instance), and completing the square – to arrive at this solution. Understanding these methods equips you with the tools to solve a wide variety of quadratic equations encountered in various mathematical and real-world contexts. Remember to always check your solutions by substituting them back into the original equation to ensure accuracy. The ability to choose the most appropriate method based on the equation's characteristics is key to efficient and accurate problem-solving.