Problem 63 - Algebra 2

Question

Factor completely. $$ x^{2}-18 x+81 $$

Step-by-Step Solution

Answer

The expression factors to \((x-9)^2\).

Step 1: Identify the quadratic form

The expression given is a quadratic polynomial, which has the form \(ax^2 + bx + c\). In this case, \(a = 1\), \(b = -18\), and \(c = 81\).

Step 2: Use the perfect square trinomial method

Notice that the polynomial \(x^2 - 18x + 81\) resembles the form of a perfect square trinomial, \((x - n)^2 = x^2 - 2nx + n^2\). Compare \(x^2 - 18x + 81\) to \(x^2 - 2nx + n^2\) to find \(n\). We see that \(-2n = -18\) and \(n^2 = 81\).

Step 3: Solve for \(n\)

From \(-2n = -18\), solve for \(n\) by dividing both sides by -2.\(-18 \div -2 = 9\). Check by using \(n^2 = 81\), as \(9^2 = 81\). Thus, \(n = 9\).

Step 4: Rewrite the expression as a perfect square

Substitute \(n = 9\) back into the form \((x - n)^2\). The factored form is \((x - 9)^2\).

Step 5: Verify the factored form

Expand \((x - 9)^2\) to check: \((x - 9)(x - 9) = x^2 - 9x - 9x + 81 = x^2 - 18x + 81\). The original expression is the same as the expanded form, confirming our factorization.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be factored into a binomial squared. This means it takes the form

• \((x + a)^2\) becoming \(x^2 + 2ax + a^2\)
• or \((x - a)^2\), which becomes \(x^2 - 2ax + a^2\).

Recognizing a perfect square trinomial can simplify the process of factoring. It essentially involves identifying two key patterns:

• The middle term is twice the product of a number and the variable \(x\).
• The last term is the square of this number.

For instance, in our exercise, the expression \(x^2 - 18x + 81\) fits the pattern

• with \(-2n = -18\), solving for \(n\) gives us \(n = 9\), and
• \(n^2 = 81\), which checks out.

This match confirms that \(x^2 - 18x + 81\) is a perfect square trinomial, allowing us to factor it easily as \((x - 9)^2\). This method streamlines factoring quadratics by recognizing squared differences.

Quadratic Polynomials

Quadratic polynomials are expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not zero. These expressions are significant in algebra because they form the basis of many mathematical problems and solutions. Quadratic expressions often lead to parabolas when graphed.A crucial aspect of working with quadratic polynomials is understanding how to factor them, find their roots, or complete their square form. The process generally involves:

• Identifying the coefficients \(a\), \(b\), and \(c\).
• Recognizing patterns such as perfect square trinomials or using methods like completing the square or applying the quadratic formula.

The quadratic polynomial in our example, \(x^2 - 18x + 81\), shows one way these patterns can help simplify polynomial expressions, leading to more straightforward solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the building blocks of algebra and can represent a wide range of mathematical situations. In algebra, expressions like

• monomials (single terms)
• binomials (two terms)
• trinomials (three terms)

are common, each characterized by the number of terms they contain. Simplifying algebraic expressions is about combining like terms and applying rules to rewrite expressions in simpler or more useful forms. Factoring, a key skill in algebra, is essential for simplifying products of expressions and solving equations. Understanding how to factor expressions, such as converting a trinomial like \(x^2 - 18x + 81\) into \((x - 9)^2\), highlights how algebraic manipulation can reveal underlying patterns and structures, leading to elegant and efficient solutions.