Problem 45 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. $$ y^{2}+y $$

Step-by-Step Solution

Answer

Add \( \frac{1}{4} \) to make the trinomial a perfect square.

Step 1: Identify the Binomial

The binomial given is \( y^2 + y \). We need to find a constant to add to this binomial so that it becomes a perfect square trinomial.

Step 2: Recall the Perfect Square Formula

The general form of a perfect square trinomial is \( (a + b)^2 = a^2 + 2ab + b^2 \). Here, the trinomial \( y^2 + 2by + b^2 \) will be a perfect square of the form \( (y + b)^2 \).

Step 3: Solve for the Missing Constant

For the trinomial \( y^2 + y + c \) to be a perfect square, the middle term \( y \) must be written as \( 2by \). Set \( 2b = 1 \) to solve for \( b \). Dividing both sides by 2 yields \( b = \frac{1}{2} \).

Step 4: Calculate the Value of the Constant

Substitute \( b = \frac{1}{2} \) into \( b^2 \) to find the constant term. \( b^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). This is the constant that needs to be added to the binomial.

Step 5: Write the Perfect Square Trinomial

The trinomial becomes \( y^2 + y + \frac{1}{4} \), which is a perfect square of \( \left(y + \frac{1}{2}\right)^2 \).

Key Concepts

Perfect Square Formula

The perfect square formula is a vital tool in algebra, especially when dealing with polynomials. This formula allows us to represent a binomial squared as a trinomial. The standard formula for this is given by

• \((a + b)^2 = a^2 + 2ab + b^2\)

By applying this formula, we can convert a binomial into a three-term expression (trinomial) with specific characteristics. In the context of the exercise, the goal is to adjust the binomial such that it neatly fits this formula's pattern.
One savvy move is to understand that the middle term in the trinomial represents twice the product of the two numbers in the binomial. Recognizing this pattern makes identifying and calculating any missing constants easier.

Binomial

A binomial is a mathematical expression containing two terms. Typically, it's written in the form:

• \(a + b\)

In this case, our binomial is \(y^2 + y\).
Binomials are often the starting point for forming more complex algebraic expressions such as trinomials. Understanding the role of each part in the binomial is essential for manipulating it, particularly when you aim to create a perfect square trinomial.
Remember that when dealing with perfect squares, the binomial terms will be transformed following the perfect square formula into a specific three-term expression.

Trinomial

A trinomial consists of three terms. When derived from a binomial squared, it often follows the pattern:

• \(a^2 + 2ab + b^2\)

The task in the exercise was to convert the binomial \(y^2 + y\) into a perfect square trinomial by adding an appropriate constant term. This usually results in a trinomial that can be expressed as the square of a binomial, like \((y + b)^2\).
A perfect square trinomial is special because it represents the expanded form of a squared binomial, which makes it perfect for simplification and solving quadratic equations. The added constant term in the process is crucial for maintaining equality and balance in the transformation.

Constant Term

The constant term in a trinomial is critical when turning a simple binomial into a perfect square trinomial. From the step-by-step solution, we found that the constant term completes the square. It is represented as \(b^2\) in the formula \((a + b)^2 = a^2 + 2ab + b^2\).
In the exercise, for the binomial \(y^2 + y\), by setting the middle term \(2by\) to match the given single \(y\), we solve for \(b\), discovering it to be \(\frac{1}{2}\).
Next, we calculate the constant term as \(b^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).
This constant term \(\frac{1}{4}\) balances our trinomial \(y^2 + y + \frac{1}{4}\) into a perfect square form, beautifully fitting the expression \((y + \frac{1}{2})^2\). Recognizing how to identify and use the constant term is a fundamental skill in algebraic transformations.