Question

Use the technique of completing the square to express each trinomial as the square of a binomial. $$ 4 x^{2}-4 x+1 $$

Short answer

The trinomial is \((2x - 1)^2\).

Step-by-step solution

Identify and Factor Out Coefficient of Quadratic Term

The given trinomial is \(4x^2 - 4x + 1\). Notice that the quadratic term has a coefficient of 4. Start by factoring 4 out of the first two terms: \[4(x^2 - x) + 1\].

Complete the Square Inside the Parenthesis

Focus on the expression inside the parenthesis, \(x^2 - x\). To complete the square, take half of the linear coefficient (-1), square it, and add inside the parenthesis. Half of -1 is \(-\frac{1}{2}\), and its square is \(\frac{1}{4}\). Adjust accordingly: \[4\left(x^2 - x + \frac{1}{4} - \frac{1}{4}\right) + 1\].

Rearrange and Simplify the Expression

Add and subtract \(\frac{1}{4}\) inside the bracket, allowing us to complete the square: \[4\left((x - \frac{1}{2})^2 - \frac{1}{4}\right) + 1\].

Simplify and Write as a Single Expression

Distribute the 4 through the completed square expression: \[4(x - \frac{1}{2})^2 - 1 + 1\]. The \(-1\) and \(+1\) cancel out, so we are left with \[4(x - \frac{1}{2})^2\].

Write the Final Expression as a Square of a Binomial

The trinomial \(4x^2 - 4x + 1\) is expressed as the square of a binomial: \((2x - 1)^2\).

Key concepts

Quadratic Expressions

Quadratic expressions are mathematical expressions where the highest exponent of a variable is 2. They typically take the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our specific example, the expression \(4x^2 - 4x + 1\) fits this structure, with \(a = 4\), \(b = -4\), and \(c = 1\). Understanding the structure of a quadratic expression is crucial because it allows us to apply certain methods for simplification, such as factoring or completing the square. Quadratic expressions are often encountered in various fields like physics, engineering, and finance, where they help model parabolic trends. They can represent anything from projectile motion to calculating interest rates. To solve or manipulate these expressions, we frequently turn to techniques like factoring or using the quadratic formula. That said, 'completing the square' is one versatile method particularly useful for rewriting quadratic expressions in a more manageable form.

Binomial Squares

Binomial squares arise when you take a binomial expression, such as \((a + b)\) or \((a - b)\), and square it. The squared binomial results in a trinomial expressed as \((a + b)^2 = a^2 + 2ab + b^2\) or \((a - b)^2 = a^2 - 2ab + b^2\). In our exercise, the trinomial \(4x^2 - 4x + 1\) was transformed into the square of a binomial, specifically \((2x - 1)^2\).
The process of finding a binomial square from a trinomial involves completing the square. This technique adds and subtracts the necessary terms to create a perfect square trinomial, making it easier to express as a binomial square. Completing the square is not only useful for converting quadratic expressions; it's essential in algebra for solving quadratic equations as well. By rearranging our initial expression into a binomial square, it becomes more straightforward to analyze its properties, such as its root, vertex, and symmetry in graphing scenarios.

Factoring Quadratic Equations

Factoring quadratic equations is a method used to break down complex quadratic expressions into products of simpler binomial expressions. This technique is immensely helpful not only for simplifying expressions but also for solving them by finding their roots. In the context of our example, after completing the square, the trinomial \(4x^2 - 4x + 1\) was rewritten as \((2x - 1)^2\). Factoring allowed us to identify the root directly, which is \(x = \frac{1}{2}\).
To factor a quadratic equation, you often look for numbers that can multiply to give you \(ac\) (the product of the leading and constant terms) and also add up to \(b\) (the linear coefficient). However, not all quadratic equations are easy to factor by inspection alone, which is why completing the square or using the quadratic formula are also valuable tools. Factoring is central in algebra due to its utility in solving not just quadratic equations but also higher-degree polynomials. Understanding this concept ensures the ability to approach not only homework problems with confidence but also real-world mathematical modeling challenges.