Problem 87 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(4 x^{2}-12 x y+9 y^{2}\)

Step-by-Step Solution

Answer

The expression is factored as \((2x - 3y)^2\).

Step 1: Recognize the Form

The expression given, \(4x^2 - 12xy + 9y^2\), is a trinomial. Notice that it resembles the quadratic form \(ax^2 + bxy + cy^2\). We will check if it is a perfect square trinomial.

Step 2: Check for Perfect Square

A perfect square trinomial takes the form \((mx + ny)^2 = m^2x^2 + 2mnxy + n^2y^2\). Here, \(m^2 = 4\), \(2mn = -12\), and \(n^2 = 9\). We will solve these to identify potential values for \(m\) and \(n\).

Step 3: Solve for m and n

Start with \(m^2 = 4\), which gives \(m = 2\) or \(m = -2\). Then for \(n^2 = 9\), we get \(n = 3\) or \(n = -3\). Check for consistency with \(2mn = -12\). Substituting known values: 2(2)(3) = 12 does not match, but 2(-2)(3) does (-12). This implies \(m = 2\), \(n = -3\).

Step 4: Factor the Expression

Since \(m = 2\) and \(n = -3\) satisfy the equation for perfect squares, we factor \(4x^2 - 12xy + 9y^2\) as \((2x - 3y)^2\).

Step 5: Verify the Solution

To ensure the factoring is correct, expand \((2x - 3y)^2\) to check if it results in the original expression. It expands to \((2x)^2 - 2 \cdot 2x \cdot 3y + (-3y)^2 = 4x^2 - 12xy + 9y^2\), confirming our factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial. It can be expressed in the form

• \((mx + ny)^2\), where "m" and "n" are constants.
• Expanded, it will look like this: \(m^2x^2 + 2mnxy + n^2y^2\).

The defining feature of a perfect square trinomial is that it factors into a binomial squared. This means you can easily find its roots by identifying which square and square root terms box it between them.Let’s look at an example:

• Consider the trinomial \(4x^2 - 12xy + 9y^2\).
• We can see it fits the perfect square pattern because \(m^2 = 4\), \(2mn = -12\), and \(n^2 = 9\).

This reveals a symmetry and balance in the trinomial structure, indicating its perfect square nature. Hence, the trinomial can be rewritten as \((2x - 3y)^2\), since the conditions for a perfect square trinomial are satisfied.

Quadratic Expression

A quadratic expression is a polynomial that involves terms up to the second degree. Its general form is

• \(ax^2 + bxy + cy^2\), where "a", "b", and "c" are coefficients.

In the exercise, we start with the quadratic expression \(4x^2 - 12xy + 9y^2\). Here:

• The term \(4x^2\) is the quadratic term associated with \(x\).
• The term \(-12xy\) represents the interaction between "x" and "y", a mixed term.
• Lastly, \(9y^2\) is the quadratic term associated with \(y\).

Quadratic expressions can often be factored, simplified, or addressed in various mathematical contexts such as in solving quadratic equations, optimizing questions in calculus, or analyzing parabolic paths in physics. Recognizing a quadratic form is the first step in performing these operations, as it sets the foundation for deeper exploration into the behavior of the expression.

Algebraic Factoring

Algebraic factoring involves breaking down expressions into simpler ones that can be multiplied to recreate the original expression. With practice, spotting opportunities for factoring becomes intuitive.For a trinomial, the ideal factoring simplifies it into binomial expressions. In our case, we identified \((2x - 3y)^2\) as the factored form of \(4x^2 - 12xy + 9y^2\). Here's how factoring works:

• Identify patterns or special forms (like perfect squares) in the expression.
• Apply corresponding factorizations.
• Use simple arithmetic to solve for necessary coefficients.
• Verify the factorization by expanding it back out to check if it matches the original expression.

Algebraic factoring is a powerful tool because it simplifies problems, makes solving equations easier, and helps in understanding mathematical models better. It’s like reverse-engineering the expression, allowing for clearer insight and potential problem-solving paths.