Problem 86 - A Graphical Approach to College Algebra

Question

Factor by any method. $$(3 a+5)^{2}-18(3 a+5)+81$$

Step-by-Step Solution

Answer

\((3a - 4)^2\)

Step 1: Identify Substitution Variable

Identify a substitution variable to simplify the expression. Let \( x = 3a + 5 \). This will transform the expression to \( x^2 - 18x + 81 \).

Step 2: Rewrite the Expression

Rewrite the expression using the substitution. The expression \((3a + 5)^2 - 18(3a + 5) + 81 \) becomes \( x^2 - 18x + 81 \).

Step 3: Recognize Perfect Square

Notice that \( x^2 - 18x + 81 \) is a perfect square trinomial. It follows the form \((x - n)^2\) where \( n^2 = 81 \) and \( 2nx = 18x \).

Step 4: Factor the Perfect Square

Factor \( x^2 - 18x + 81 \) as \((x - 9)^2\) because \(9^2 = 81\) and \(2 \cdot 9 \cdot x = 18x\).

Step 5: Substitute Back

Substitute \( x = 3a + 5 \) back into the factored expression. The equation \((x - 9)^2\) becomes \((3a + 5 - 9)^2\).

Step 6: Simplify Substituted Expression

Simplify \((3a + 5 - 9)^2\) to get \((3a - 4)^2\).

Step 7: Verify the Solution

Verify the factored form by expanding \((3a - 4)^2\) to ensure it matches the original expression.

Key Concepts

Substitution Method

The substitution method is a useful algebraic technique that simplifies complex expressions, making them easier to factor or solve. In this method, we choose a substitution variable to replace part of the expression. This turns the expression into a simpler form.

In the given exercise, the expression \[(3a + 5)^2 - 18(3a + 5) + 81\]The substitution variable is chosen as \(x = 3a + 5\).

• This transforms the expression into \(x^2 - 18x + 81\).
• Substitution allows you to focus on factoring a more familiar trinomial.

By substituting, you simplify the process, making it easier to recognize patterns like a perfect square trinomial later on.

After factoring, remember to substitute back to find the solution in terms of the original variable. This method is powerful because it breaks down the problem into smaller, more manageable parts.

Trinomial Factoring

Trinomial factoring involves expressing a quadratic expression as a product of two binomials. This is a core technique in algebra when dealing with polynomials of the form \(ax^2 + bx + c\).

For the expression \(x^2 - 18x + 81\), it fits the form \(ax^2 + bx + c\) where \(a = 1\), \(b = -18\), and \(c = 81\).

• Factors of \(c\) that add up to \(b\) determine the terms of the binomials.
• For this trinomial, both factors are \(9\) (since \(81 = 9 imes 9\) and \(-9 + -9 = -18\)).

Recognizing special patterns like the perfect square trinomial can speed up this process. Here, \(x^2 - 18x + 81\) is a perfect square trinomial, which simplifies factoring.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form \((u + v)^2 = u^2 + 2uv + v^2\).

In our example, the expression after substitution is \(x^2 - 18x + 81\). This is a perfect square trinomial because:

• It resembles \((x - n)^2\) form with \(n = 9\).
• We notice that \(81 = 9^2\) and the middle term, \(-18x\), matches \(-2 \times 9 \times x\).

Therefore, it factors neatly into \((x - 9)^2\).

Recognizing a perfect square trinomial helps in quickly rewriting it as a square of a binomial, saving time and reducing error. Always verify by expanding to confirm it matches the original expression.