Problem 84 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ 4 x^{2}-12 x y+9 y^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the given trinomial is $$(2x - 3y)^2$$.

Step 1: 1. Identify the trinomial form

The given trinomial is $$4x^2 - 12xy + 9y^2$$. We can observe that this is a quadratic of the form $$ax^2 + 2abxy + b^2y^2$$, which can be factored using the perfect square trinomial formula.

Step 2: 2. Check if the trinomial is a perfect square

To check if the trinomial is a perfect square, we can see if the first and last terms are perfect squares, and if the middle term is 2 times the product of their square roots. Here, the first term is $$4x^2 = (2x)^2$$, and the last term is $$9y^2 = (3y)^2$$. The middle term is $$-12xy = 2(-2x)(3y)$$. Since all conditions are met, the trinomial is indeed a perfect square.

Step 3: 3. Apply the perfect square trinomial formula

Since the trinomial is of the form $$a^2x^2 - 2abxy + b^2y^2$$, we can factor it using the formula $$(ax - by)^2$$. In this case, $$a = 2x$$ and $$b = 3y$$, so the factored form is $$(2x - 3y)^2$$.

Step 4: 4. Write the final answer

The factored form of the given trinomial $$4x^2 - 12xy + 9y^2$$ is $$(2x - 3y)^2$$.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of trinomial that takes the shape of \[a^2x^2 + 2abxy + b^2y^2\]. This structure makes it easy to factor using a specific formula.
Here’s how to recognize one:

• The first term should be a perfect square, meaning something like \(a^2\) where \(a\) is a whole number or a variable.
• Similarly, the last term also needs to be a perfect square, such as \(b^2\).
• The middle term is crucial; it must be twice the product of the square roots of the first and last terms, written as \(2abxy\).

In our example trinomial, \[4x^2 - 12xy + 9y^2\], we can see that:

• \(4x^2\) is a perfect square \( (2x)^2\).
• \(9y^2\) is also a perfect square \((3y)^2\).
• The middle term, \(-12xy\), fits because it equals \(2(-2x)(3y)\).

These checks confirm it's a perfect square trinomial.

Quadratic Equations

A quadratic equation is any equation that can be rewritten in the standard form \[ax^2 + bx + c = 0\].
Quadratics frequently appear in various forms, such as trinomials. But not all trinomials are quadratic equations.

In the context of the perfect square trinomial \[4x^2 - 12xy + 9y^2\]:

• Here, \(a = 4\), \(b = -12y\), and \(c = 9y^2\).
• It's a quadratic in terms of the variable \(x\).

Quadratic equations can be factored, solved by completing the square, or using the quadratic formula.
Identifying a perfect square trinomial simplifies the factoring process as it allows us to use the formula of a perfect square directly.

Factoring Techniques

Factoring is the process of breaking down an expression into products, making it easier to solve or simplify.
For a trinomial like \[4x^2 - 12xy + 9y^2\], factoring involves finding the expression set to a product of binomials.

Here, the technique is to identify a special pattern like a perfect square.

• Recognizing \( (2x)\) and \( (3y)\) allows us to quickly factor the trinomial into \((2x - 3y)^2\).
• This technique is a time-saver when conditions point to a perfect square trinomial.

Other general factoring techniques include:

• Factoring by grouping, useful when no special pattern is immediately apparent.
• Using formulas like the difference of squares for specific forms.

Mastering these techniques aids in solving quadratic equations and simplifies complex algebraic expressions.