Problem 12 - Elementary Algebra

Question

Factor. $$ 4 x 2-20 x+25 $$

Step-by-Step Solution

Answer

The expression factors to \((2x - 5)^2\).

Step 1: Identify the Coefficients

First, identify the coefficients of each term in the quadratic expression. For the expression \(4x^2 - 20x + 25\), the coefficients are 4, -20, and 25.

Step 2: Check for a Perfect Square Trinomial

Recognize that the given expression might be a perfect square trinomial. A perfect square trinomial can be expressed as \((ax \, + \, b)^2\). Identify if the form \((a^2x^2 - 2abx + b^2)\) matches \(4x^2 - 20x + 25\).

Step 3: Calculate Values for 'a' and 'b'

Set the conditions to check the perfect square trinomial: \(a^2 = 4\), which gives \(a = 2\); \(b^2 = 25\), which gives \(b = 5\); and \(-2ab = -20\), which confirms \(-2(2)(5) = -20\).

Step 4: Write the Expression as a Binomial Square

Since all conditions for a perfect square trinomial are satisfied, write the expression as a square of a binomial: \((2x - 5)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression that can be written as the square of a binomial.
In simpler terms, it is a trinomial that originated from multiplying a binomial by itself. This means that expressions like \(a^2x^2 - 2abx + b^2\) can be neatly transformed into the form \( (ax + b)^2\).
Recognizing a perfect square trinomial involves three steps:

• Finding a term in the quadratic expression that is a perfect square.
• Ensuring that the middle term is twice the product of the square root of this term and another term.
• Checking if the last term is a perfect square.

Perfect square trinomials make factoring much simpler because they allow the expression to be rewritten as a squared binomial. This helps in solving quadratic equations swiftly.

Coefficients

In a quadratic expression, coefficients are the numbers that multiply the variables.
For the expression \( 4x^2 - 20x + 25 \), the coefficients are 4, -20, and 25.
Understanding coefficients is crucial when factoring because they guide us to identify patterns like perfect square trinomials.

• The coefficient of \(x^2\) here is 4.
• The coefficient of \(x\) is -20.
• The constant term, which can be seen as a coefficient for \(x^0\), is 25.

When factoring, these numbers help us find suitable 'a' and 'b' values that satisfy the form \(a^2x^2 - 2abx + b^2\). Correct identification and manipulation of coefficients are key to successful factoring.

Binomial Square

The concept of a binomial square is closely tied to perfect square trinomials. A binomial square is what you get when you multiply a binomial by itself.
In simple terms, a binomial square looks like this: \( (ax + b)^2 \).When you expand this form, using the distributive property, it becomes \(a^2x^2 + 2abx + b^2\).

• Each binomial square results in a perfect square trinomial.
• This structure is essential when seeing if a quadratic expression can be factored as a binomial square.

Recognizing a binomial square simplifies the factoring process because it provides a direct method to write a quadratic expression as \( (ax + b)^2 \). This transformation is not just neater, but also reveals the roots or solutions of the quadratic equation when needed.

Quadratic Expression

A quadratic expression is an algebraic expression of the form \( ax^2 + bx + c \). It involves a variable raised to the second power, hence the name 'quadratic.'
The understanding of quadratic expressions is vital in algebra as they form the foundation for understanding more complex equations.

• The term \(ax^2\) is called the quadratic term.
• The \(bx\) term is the linear term, and it dictates the slope of the quadratic.
• The constant term \(c\) shifts the graph of the equation up or down the y-axis.

Factoring quadratic expressions often involve checking for patterns like perfect square trinomials or using methods like the quadratic formula. Grasping the structure of a quadratic expression enables one to apply various strategies to find its factors or solutions efficiently, providing insights into key properties of the equation.