Problem 4 - Introductory Algebra for College Students

Question

Complete the square for binomial. Then factor the resulting perfect square trinomial. \(x^{2}-4 x\)

Step-by-Step Solution

Answer

The factored form of the given binomial \(x^{2}-4 x\) after completing the square is \((x - 2)^2\).

Step 1: Rewrite the binomial

The given binomial is \(x^{2}-4x\). Rewriting this gives us \(x^2 - 4x + o\).

Step 2: Complete the square

To turn the trinomial into a square of a binomial, the third term should be \((\frac{b}{2})^2\). Given the second term -4x, we have \(-4/2 = -2\) as our 'b'. This gives the term \(-2^2 = 4\) which we add to the binomial, transforming it into a perfect square trinomial: \(x^2 - 4x + 4\).

Step 3: Factor the perfect square trinomial

We can now factor the perfect square trinomial. It will be \((a - b)^2\). In this case, 'a' is 'x' and 'b' is 2. Therefore the expression becomes: \((x - 2)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of algebraic expression that can be rewritten as the square of a binomial. It's an essential concept when completing the square, a common method for solving quadratic equations. To tell if a trinomial is a perfect square:

• It needs to have three terms.
• The first and third terms should be perfect squares themselves.
• The middle term should be twice the product of the square roots of the first and third terms.

For example, in the expression \(x^2 - 4x + 4\), the first term \(x^2\) and the third term \(4\) are perfect squares, and the middle term \(-4x\) fits as \(-2 \cdot x \cdot 2\). Therefore, this trinomial is indeed a perfect square, namely \((x - 2)^2\). Understanding this pattern makes it easier to identify and create perfect square trinomials during the process of completing the square.

Factoring

Factoring is a crucial skill in algebra that involves breaking down complex expressions into simpler, multiplied factors. When dealing with a perfect square trinomial, factoring becomes straightforward if you recognize its structure.

Consider the trinomial \(x^2 - 4x + 4\), a perfect square trinomial. We established that it could be expressed as \((x - 2)\) squared. Here's why:

• Take the square root of the first term. In this case, \(\sqrt{x^2}\) is \(x\).
• Take the constant value which made it a perfect square, here it's 2.
• Combine these into \((x - 2)\), because it matches the structure \((a - b)^2\).

This conversion of trinomial back into its binomial factors simplifies solving equations. It's especially useful when you're tasked to solve quadratic equations by finding their roots.

Algebraic Expressions

Algebraic expressions, like \(x^2 - 4x\), are combinations of numbers, variables, and arithmetic operations. These expressions form the backbone of solving equations and understanding functions in algebra.

When working with algebraic expressions, especially quadratics, we often use techniques like completing the square as seen in our example. This method helps simplify expressions and thus makes solving equations with them easier.

• Understanding different components like the variable term \(x^2\) (representing some unknown value squared) and linear term \(-4x\) is essential.
• Adding a constant, in our example 4, allows forming a complete trinomial which can then be transformed into a squared binomial.

Grasping how to manipulate algebraic expressions is a key step in mastering further topics like solving quadratic equations, graphing parabolas, and exploring functions.