Problem 42 - Algebra A Combined Function

Question

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. $$ y^{2}+4 y $$

Step-by-Step Solution

Answer

Add 4 to make it a perfect square trinomial.

Step 1: Identify the coefficient of the linear term

The given expression is \(y^2 + 4y\). Here, the linear term is \(4y\), so the coefficient of \(y\) is 4.

Step 2: Divide the coefficient by 2

Take the coefficient of \(y\), which is 4, and divide it by 2: \(\frac{4}{2} = 2\).

Step 3: Square the result from Step 2

Square the result obtained in Step 2: \(2^2 = 4\). This is the constant that, when added to the binomial, makes it a perfect square trinomial.

Step 4: Form the perfect square trinomial

Add the constant found in Step 3 to the binomial \(y^2 + 4y\), resulting in \(y^2 + 4y + 4\). This expression is a perfect square trinomial.

Step 5: Verify by factoring

Verify that the expression formed is a perfect square trinomial by factoring: \(y^2 + 4y + 4\) can be factored as \((y + 2)^2\), confirming it's a perfect square trinomial.

Key Concepts

Factoring Trinomials

Factoring trinomials is an important algebraic skill that involves rewriting a quadratic expression as a product of two binomials. This is useful because it can simplify expressions and solve equations more easily.
To factor a trinomial like a perfect square trinomial, you have to recognize its form and apply the correct technique. A perfect square trinomial takes the form \( a^2 + 2ab + b^2 \) or \( a^2 - 2ab + b^2 \), which factors into \((a + b)^2\) or \((a - b)^2\), respectively.
In our example, the expression \( y^2 + 4y + 4 \) can be rewritten as \((y + 2)^2\). Here, \(y\) is the first term and \(2\) is the number squared to produce \(4\), which is the added constant.

Completing the Square

Completing the square is a technique used to form a perfect square trinomial from a quadratic expression. This method is particularly helpful in solving quadratic equations and transforming functions for graphing.
To complete the square for a quadratic expression like \( y^2 + 4y \), you follow these simple steps, similar to those in the solution:

• Identify the linear term's coefficient; here, it's \(4\)
• Divide it by \(2\), yielding \(2\)
• Square the result; \(2^2\) equals \(4\)

This \(4\) is added to the expression to complete the square, turning \( y^2 + 4y \) into \( y^2 + 4y + 4 \). The expression is now a perfect square trinomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like addition and subtraction. Understanding them is crucial because they form the foundation of algebra.
The exercise involves turning a binomial, which is a two-term expression, \( y^2 + 4y \), into a trinomial by adding a constant. This kind of manipulation helps in solving and understanding equations better.
Algebraic expressions can be simple, like \( x + 2 \), or complex, like \( 3x^2 + 7x - 5 \). By mastering how to factor and complete squares, you can handle these expressions effectively in various mathematical contexts. This will be useful across disciplines that rely on algebraic reasoning, such as physics and engineering.