Introduction to Quadratic Inequalities

Quadratic inequalities are a significant part of algebra, often appearing in various real-world applications. They involve solving inequalities where the variable has an exponent of two, such as squaring a variable. In this guide, we will delve into a specific example while optimizing for SEO and providing valuable information relevant to both students and professionals dealing with algebraic problems.

Understanding the Quadratic Equation

The given problem involves solving the quadratic equation 2x^2 - 4x - 2 14x. Let's break down the equation and solve it step by step.

Step 1: Simplify the Equation

First, we need to simplify the equation to put it in standard form. Start by moving all terms to one side of the equation:

2x^2 - 4x - 2 - 14x 0

This simplifies to:

2x^2 - 18x - 2 0

Step 2: Apply the Quadratic Formula

The general form of a quadratic equation is ax^2 bx c 0. For our equation, a 2, b -18, and c -2. We can use the quadratic formula to solve for x:

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

Evaluating the values:

[x frac{-(-18) pm sqrt{(-18)^2 - 4(2)(-2)}}{2(2)}]

[x frac{18 pm sqrt{324 16}}{4}]

[x frac{18 pm sqrt{340}}{4}]

[x frac{18 pm 2sqrt{85}}{4}]

[x frac{9 pm sqrt{85}}{2}]

Step 3: Interpret the Solutions

The solutions to our equation are:

x (frac{9 sqrt{85}}{2}) or x (frac{9 - sqrt{85}}{2})

These solutions represent the critical points where the quadratic expression changes its behavior (from greater than to less than zero or vice versa).

SEO Optimization for Quadratic Inequalities

Optimizing for Google involves ensuring that the content is not only accurate and comprehensive but also easily discoverable. Include key terms and phrases to improve search visibility.

Keyword Focus

Keywords: quadratic inequalities, solving quadratic equations, algebraic solutions

Conclusion

Understanding and solving quadratic inequalities is crucial for advancing algebraic skills and problem-solving abilities. By practicing with real-world problems and optimizing content for SEO, you can enhance your learning experience and improve your proficiency in mathematics.

Additional Resources

To further enhance your knowledge and skills, explore additional resources such as online tutorials, practice problems, and advanced reading materials on algebra and quadratic equations. Regular practice and exposure to diverse problems will significantly improve your understanding and problem-solving skills in this area.