Problem 49 - Elementary Algebra

Question

Factor. $$ x 2 y 2-6 x y+9 $$

Step-by-Step Solution

Answer

The expression factors to \((xy - 3)^2\).

Step 1: Identify the Expression

The expression given is \(x^2y^2 - 6xy + 9\). To begin factoring, recognize that it resembles a quadratic in terms of \(xy\).

Step 2: Rewrite the Expression

Think of \(xy\) as a single variable, say \(z\). This means the expression can be rewritten as \(z^2 - 6z + 9\).

Step 3: Factor the Quadratic Expression

The expression \(z^2 - 6z + 9\) is a perfect square trinomial. It can be factored as \((z - 3)^2\), because \(-3\) is a root repeated twice.

Step 4: Substitute Back the Original Variables

Now, substitute back \(xy\) for \(z\), so \((z - 3)^2\) becomes \((xy - 3)^2\).

Step 5: Verify the Factorization

Expand \((xy - 3)^2\) to verify it matches the original expression: \((xy - 3)(xy - 3) = x^2y^2 - 6xy + 9\). This confirms the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that takes the form \(a^2 - 2ab + b^2\), which can be factored into \((a-b)^2\). In the problem you solved, the expression \(z^2 - 6z + 9\) is a perfect square trinomial. Let's understand why.
\(z^2 - 6z + 9\) can be broken down into a formula where:

• \(a = z\)
• \(b = 3\)

Because the middle term, \(-6z\), fits \(-2ab\) where \(2 imes z imes 3 = 6z\), we can conclude this trinomial is a perfect square.
Thus, it factors neatly into \((z-3)^2\). This factorization is vital in simplifying many quadratic expressions, making calculations and problem-solving much easier.

Substitution Method

The substitution method is a handy mathematical technique, especially in simplifying expressions and solving equations. By temporarily replacing variables, it allows manipulation of the formula in a simpler form.
In this exercise, the term \(xy\) was replaced by a single variable, \(z\), turning the original complicated expression \(x^2y^2 - 6xy + 9\) into a friendlier form \(z^2 - 6z + 9\).
This method is particularly useful in quadratic expressions where several factors or variables interact in complex ways. This transformation helps to easily spot patterns, such as perfect square trinomials, which are less apparent in more complex expressions.
After completing the necessary algebraic operations with the substituted variable, you then revert to the original components, re-substituting \(xy\) back for \(z\). This brings the factorization full circle.

Verify Factorization

Verifying factorization is a crucial step in solving any mathematical expression, ensuring that the solution is accurate and reliable. After factoring an expression like \((xy - 3)^2\), you should always expand it back out to check whether it returns to the original equation.

This involves expanding \((xy - 3)(xy - 3)\), which calculates to:

• \(xy \times xy = x^2y^2\)
• \(-(xy) \times 3 - 3 \times (xy) = -6xy\)
• \(-3 \times -3 = 9\)

Adding these terms together brings you full circle back to \(x^2y^2 - 6xy + 9\). This step is like checking your road map to make sure you arrive back at the correct spot - it's crucial for ensuring the mathematical procedure has been executed correctly and no mistakes were made along the way.