Problem 81 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(x^{2}-12 x+36\)

Step-by-Step Solution

Answer

The expression factors as \((x - 6)^2\).

Step 1: Identify the Expression Structure

The expression given is a quadratic expression in the form of \(ax^2 + bx + c\), where \(a = 1\), \(b = -12\), and \(c = 36\).

Step 2: Check for Perfect Square Trinomial

A perfect square trinomial is of the form \((x - d)^2 = x^2 - 2dx + d^2\). Compare the expression with this form to check if it can be factored into a perfect square. Here, \(x^2 - 12x + 36\) resembles \(x^2 - 2dx + d^2\). Identify \(d\).

Step 3: Calculate the Value of \(d\)

To determine \(d\), recall that \(-2d = -12\). Solving for \(d\), divide both sides by \(-2\): \(d = 6\). Also, verify that \(d^2 = 36\) which matches \(c\).

Step 4: Write the Factored Form

Since the expression is a perfect square trinomial, we can factor it as \((x - d)^2\). Use \(d = 6\) to write the factored form: \((x - 6)^2\).

Step 5: Verify the Solution

Expand the factorized expression \((x - 6)^2\) to ensure it returns to the original expression: \((x - 6)(x - 6) = x^2 - 6x - 6x + 36 = x^2 - 12x + 36\). The original expression is recreated, confirming the correct factorization.

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that is formed by squaring a binomial. It takes the general form

• \((x + d)^2 = x^2 + 2dx + d^2\)
• or, \((x - d)^2 = x^2 - 2dx + d^2\)

This structure makes it easily recognizable and factorable into a single binomial squared.
In our exercise, we see the expression \(x^2 - 12x + 36\). Here, we need to identify if it matches the form of \((x - d)^2\).
To do this, compare the middle term and the constant term to \(-2dx\) and \(d^2\) respectively.
By setting \(-2d = -12\), we find that \(d = 6\). Also, since \(d^2 = 36\), it confirms the perfect square trinomial form as \((x - 6)^2\).
This allows us to factor the expression into a neat square, making calculations and solutions quite straightforward.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the second degree, often encountered in polynomial equations. They generally appear in the form

• \(ax^2 + bx + c\)

where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The "quadratic" aspect refers to the term \(ax^2\).
In our specific problem, the quadratic expression is \(x^2 - 12x + 36\), where \(a = 1\), \(b = -12\), and \(c = 36\), making it a standard quadratic layout.
The importance of understanding quadratic expressions lies in their wide application in solving equations, modeling curves, and finding roots. They can often be simplified by factoring, using the quadratic formula, or completing the square, each technique offering insight into the solutions and behavior of the expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the basis of algebra, a core component of mathematics, allowing us to

• generalize arithmetic operations
• solve equations
• and demonstrate relationships between quantities.

An algebraic expression could be as simple as \(3x + 2\) or more complex like a quadratic expression \(x^2 - 12x + 36\).
These expressions, when manipulated correctly, hold the key to solving algebraic problems efficiently.
By identifying the type of algebraic expression you are dealing with, whether linear, quadratic, or polynomial, appropriate factoring or solving techniques can be chosen.
In this exercise, recognizing that the given expression can be considered a quadratic expression simplifies the factoring process and helps us quickly arrive at the solution, \((x - 6)^2\).
Grasping the underlying principles of algebraic expressions speeds up and deepens understanding in mathematics.