How to Factor the Expression 4x^2 12xy ay^2

Mathematics is a fundamental discipline, and polynomials are at the heart of algebra. Understanding how to factor expressions is crucial for various applications, from simplifying equations to solving complex problems. In this guide, we will explore how to factor the quadratic expression 4x^2 12xy ay^2. We'll delve into the methods available and provide a step-by-step walkthrough, making it clear and accessible for students and professionals alike.

Introduction to Factoring Quadratic Expressions

Factoring, or factoring out, means to break down an expression into simpler polynomials that, when multiplied together, give the original expression. The expression 4x^2 12xy ay^2 can be factored under certain conditions, and this process can be managed through various techniques, which we will discuss in detail.

Factoring by Pulling Out Common Factors

One simple method is to factor out the greatest common factor (GCF) from the expression. Here, we can pull out 4 from the first two terms:

4x^2 12xy ay^2 4(x^2 3xy) ay^2

However, this leaves the term ay^2 still in place. Alternatively, we can factor out y from the last two terms, as below:

4x^2 12xy ay^2 4x^2 y(12x ay)

These steps simplify the expression, but they might not provide a complete factorization depending on the values of x and y.

Complete Factorization Using the Quadratic Formula

For a complete factorization, we can use the quadratic formula. When faced with a quadratic expression in the form ax^2 bx c, the quadratic formula is given by:

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

In our case, we treat the expression 4x^2 12xy ay^2 as a quadratic in x, which can be solved as follows:

4x^2 (12y)x ay^2 0

Here, a 4, b 12y, and c ay^2. Applying the quadratic formula, we get:

x frac{-12y pm sqrt{(12y)^2 - 4(4)(ay^2)}}{2(4)}

This simplifies to:

x frac{-12y pm sqrt{144y^2 - 16ay^2}}{8}

Further, this becomes:

x frac{-12y pm sqrt{16y^2(9 - a)}}{8}

And finally:

x frac{-12y pm 4y sqrt{9 - a}}{8}

This can be simplified to:

x frac{-3y pm y sqrt{9 - a}}{2}

x y(-frac{3}{2} pm frac{sqrt{9 - a}}{2})

Thus, the roots of the quadratic expression are:

x frac{12 pm sqrt{144 - 16a}}{8}y

Condition for Rational Solutions

For the solutions to be rational, the discriminant – the expression under the square root – must be a perfect square. Thus, we have:

144 - 16a n^2, where n is an integer.

Solving for a, we get:

a 9 - frac{n^2}{16}

Conclusion

Factoring the expression 4x^2 12xy ay^2 involves pulling out common factors, applying the quadratic formula, and ensuring the discriminant is a perfect square to achieve rational solutions. By understanding these methods, you can master the art of factoring quadratic expressions, which is a critical skill in algebra and beyond.