Problem 66 - Elementary Algebra

Question

Factor. Either factor out the greatest common factor, factor by grouping, use the guess and check method, or use the \(a c\) method. $$ 4 y^{2}-28 y+49 $$

Step-by-Step Solution

Answer

\((2y - 7)^2\)

Step 1: Identify common factors

Examine the given expression to see if there is a greatest common factor (GCF) that can be factored out from all terms. For the expression $$4y^2 - 28y + 49$$, notice that all terms do not have a common factor other than 1, so we move to the next method.

Step 2: Recognize the form of the quadratic

Observe that the quadratic equation is in the standard form, $$ax^2 + bx + c$$, where $$a = 4$$, $$b = -28$$, and $$c = 49$$. Notice that the given quadratic appears to be a perfect square trinomial.

Step 3: Verify if it is a perfect square trinomial

Check if the first term (\ref{ 4y^2}), last term (\ref{ 49}), and middle term (\ref{-28y}) form a perfect square trinomial. \(4y^2\) is \((2y)^2\) and \(49\) is \((7)^2\) and the middle term is \(-28y\), which is 2 times the product of \(2y\) and \(7\), considering the negative sign. So, it confirms that it is a perfect square trinomial.

Step 4: Factor the perfect square trinomial

Since the quadratic is a perfect square trinomial, it can be factored as \((2y - 7)^2\).

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides all the terms in a given expression without leaving a remainder. To find the GCF of an expression, follow these steps. First, list the factors of each term.
For example, in the expression \(4y^2 - 28y + 49\), you would list the factors for each term:

• \(4y^2\) factors: 1, 2, 4, y, \(y^2\)
• \(-28y\) factors: 1, 2, 4, 7, 14, 28, y
• \(49\) factors: 1, 7, 49

If there is a common factor that appears in all lists, it's the GCF. In this case, the only common factor is 1, meaning no other common factor exists.

Understanding the GCF helps to simplify expressions by dividing each term by this common factor, making further factoring techniques easier.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic equation that can be expressed as the square of a binomial. It takes the form \(a^2 + 2ab + b^2\). Let's look at how this applies to the expression \(4y^2 - 28y + 49\).

First, recognize whether the expression fits the perfect square trinomial form:

• The first term \(4y^2\) is a square (\((2y)^2\)).
• The last term \(49\) is a square (\((7)^2\)).
• The middle term \(-28y\) is twice the product of the square roots of the first and last terms with a negative sign, \(2 \times 2y \times (-7)\).

When these conditions hold true, the expression is indeed a perfect square trinomial, allowing it to be factored as the square of a binomial. In this example, \(4y^2 - 28y + 49 = (2y - 7)^2\).

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form \(ax^2 + bx + c = 0\). Each quadratic equation can be solved using various methods such as factoring, the quadratic formula, or completing the square.

In this exercise, the quadratic equation \(4y^2 - 28y + 49\) is evaluated to determine the method of factoring. By identifying it as a perfect square trinomial, you can factor it directly.

Factoring quadratic equations often involves:

• Identifying a common factor or GCF across all terms. If no GCF exists, use other factoring methods.
• Recognizing special forms, such as perfect square trinomials, which simplify the process.
• Verifying by expanding the binomial to ensure correct factoring.

Understanding these aspects allows for more confident and accurate solving of quadratic equations, an essential skill in algebra and beyond.