Problem 61 - Elementary Algebra Within Reach

Question

In Exercises 59?62, factor the expression. $$ 4 y^{2}-20 y+25 $$

Step-by-Step Solution

Answer

The factorized form of the expression \( 4y^2-20y+25 \) is \( (-2y+5)^2 \).

Step 1: Identify the coefficients

The coefficients in the standard format \( ax^2+bx+c \) are as follows: \( a=4, b=-20, c=25 \).

Step 2: Apply perfect square trinomial rule

In a perfect square trinomial, \( b=2*a*sqrt(c)\) and \( c=(a*sqrt(c))^2\). Checking \( b=2*a*sqrt(c)\) for our coefficients, we get \( -20 = 2*4*\sqrt{25} = -40 \), which is not true, so we need to take into account that 'a' could be negative as well because squaring a negative number also results to positive. So, if \( a=-4 \), then we get \( -20 = 2*(-4)*\sqrt{25} = -20 \), which is true. So, comparing with perfect square trinomial, we get \( a=\sqrt{4} \) and \( b=\sqrt{25} \).

Step 3: Apply factoring formula

Now apply the standard formula for perfect squares, which is \( (a±b)^2 \). So, the factorised form of the expression would be \( (-2y+5)^2 \) where \( a=-2y \) and \( b=5 \).

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression. It takes the form \( (ax + b)^2 \), which expands to \( a^2x^2 + 2abx + b^2 \). This means when you see a trinomial, if it fits the pattern of a perfect square trinomial, you can rewrite it as the square of a binomial.

To identify a perfect square trinomial:

• Check that the first and last terms are perfect squares, for instance, \( a^2x^2 \) and \( b^2 \).
• Ensure the middle term is twice the product of the square roots of the first and last terms, \( 2abx \).

Recognizing a perfect square trinomial makes factoring simpler, as it can be factored into a single squared binomial.

Exploring Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two. They typically have the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The key features of quadratic expressions include:

• The term \( ax^2 \) represents the squared variable, adding curvature to the graph of the expression.
• The term \( bx \) is the linear component, influencing the position and direction of the parabola.
• The constant \( c \) is the y-intercept of the graph.

Quadratic expressions are fundamental in algebra due to their applications in various mathematical and real-world contexts. Recognizing the coefficients and understanding their roles are crucial when working with these expressions, especially when applying factoring techniques.

Applying Factoring Techniques

Factoring involves breaking down an expression into a product of simpler expressions. It's especially useful in solving quadratic equations and simplifying polynomial expressions. Here's how you can apply factoring techniques:

• Look for Patterns: Identify any recognizable patterns like perfect square trinomials or difference of squares.
• Factor by Grouping: Useful when expressions have four terms; group terms to factor out common factors.
• Use the AC Method: Multiply \( a \) and \( c \), then find two numbers that multiply to \( ac \) and add to \( b \).

In the context of factoring perfect square trinomials, once you've recognized the special form, you can directly apply the perfect square binomial formula: \((a±b)^2\). This transforms the trinomial into a binomial squared, making it manageable and elegant. Understanding these techniques empowers you to tackle a wide range of algebraic problems effectively.