Solving Quadratic Equations: A Comprehensive Guide with Google SEO Optimization

Quadratic equations are a fundamental part of algebra and are used in a wide range of applications, from physics to engineering. In this article, we will delve into the process of solving the equation (frac{2 times 9}{y - 3} y) step by step. We will cover the methods of solving quadratic equations, including cross-multiplication, factoring, and the quadratic formula. By the end of this guide, you will have a solid understanding of how to solve these equations and improve your search engine optimization (SEO) practices.

Understanding the Equation

The given equation is (frac{2 times 9}{y - 3} y). To simplify, we first rewrite it as:

(frac{18}{y - 3} y)

Solving the Equation

Our goal is to solve for y. Here's a step-by-step breakdown of the process:

1. Cross-Multiplication Method

We can eliminate the fraction by cross-multiplying. This gives us:

18 y(y - 3)

Expanding the right side, we get:

18 y2 - 3y

Rearranging the equation to standard quadratic form, we get:

y2 - 3y - 18 0

2. Factoring the Quadratic Equation

Now, we need to factor the quadratic equation. We look for two numbers that multiply to -18 and add to 3. The numbers 6 and -3 satisfy these conditions:

(y - 6)(y 3) 0

Setting each factor to zero gives the possible solutions:

y - 6 0 implies y 6

y 3 0 implies y -3

Therefore, the solutions for y are y 6 or y -3.

3. Using the Quadratic Formula

Alternatively, we can use the quadratic formula to solve the equation y2 - 3y - 18 0. The quadratic formula is given by:

y (frac{-b pm sqrt{b^2 - 4ac}}{2a})

For our equation, a 1, b -3, and c -18. Plugging these values into the formula, we get:

y (frac{-(-3) pm sqrt{(-3)^2 - 4(1)(-18)}}{2(1)})

y (frac{3 pm sqrt{9 72}}{2})

y (frac{3 pm sqrt{81}}{2})

y (frac{3 pm 9}{2})

This gives us two possible solutions:

y (frac{3 9}{2} 6)

y (frac{3 - 9}{2} -3)

Again, the solutions are y 6 and y -3.

Avoiding Division by Zero

It's important to note that y cannot be -3, as this would make the denominator zero, resulting in an undefined expression. Therefore, the only valid solution is:

y 6

Conclusion

In summary, we have shown several methods for solving the quadratic equation (frac{2 times 9}{y - 3} y). The final solutions are y 6 or y -3. Understanding these methods not only helps in solving equations but also improves your SEO by providing comprehensive and useful content to your audience. By using SEO-friendly keywords and structuring your content logically, you can enhance your visibility on search engines.