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Chapter 6: Problem 84

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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 $$(2x - 3y)^2$$.
04

4. Write the final answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

For the following problems, factor the trinomials if possible. $$ 8 a^{2}+10 a b-6 b^{2} $$

For the following problems, factor, if possible, the trinomials. $$ b^{2}-2 b+1 $$

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

Factor \(x^{5}-8 x^{4}+7 x^{3}\).

For the following problems, factor, if possible, the trinomials. $$ 25 a^{2}-20 a+4 $$
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