Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Factor the expression. $$4 y^{2}-12 y+9$$

Step-by-Step Solution

Answer

The factored form of the expression \(4y^{2} - 12y + 9\) is \((2y - 3)^{2}\).

Step 1 : Identify the values of a,b and c

In the given expression \(4y^{2} - 12y + 9\), we can identify the constants as follows: \(a = 4\), \(b = -12\), and \(c = 9\).

Step 2 : Prepare for factoring

The given expression appears to be a perfect square trinomial, which is a trinomial in the form \(a^{2} - 2ab + b^{2}\). In order to check if it's a perfect square, compare the original equation with the form \(a^{2} - 2ab + b^{2}\). So, \(a = √4 = 2\), \(b = √9 = 3\) and the equation becomes \((2y)^{2} - 2*2*3y + 3^{2}\).

Step 3 : Confirming and factoring

Confirm if the equation in step 2 is indeed true and is of the form \(a^{2} - 2ab + b^{2}\) by checking the values. (2*2y*3 = 2*2*3y). Yes, it matches. Then, factorize it as \((2y - 3)^{2}\). This is because, for a perfect square trinomial, the factored form will be \((a - b)^{2}\) where \(a = 2y\) and \(b = 3\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It takes the form \(a^{2} - 2ab + b^{2}\). When you factor this expression, it neatly breaks down into \((a-b)^{2}\). Understanding perfect square trinomials makes it easier to recognize and factor these types of equations.

For example, consider our original expression: \(4y^{2} - 12y + 9\). Notice how it can be rewritten in the perfect square trinomial format by looking for specific terms that fit into \(a^{2}, -2ab, b^{2}\).

In our case, \(4y^{2}\) is \((2y)^{2}\), and \(9\) is \(3^{2}\). Verifying the middle term, \(-12y\), fits \(-2*2y*3\). Successfully reorganizing the expression indicates it's indeed a perfect square trinomial.

Identifying Coefficients

Identifying coefficients is an essential step when working with quadratic expressions. These coefficients are the numerical values in front of each term in a quadratic equation of the form \(ax^{2} + bx + c\).

In our quadratic equation \(4y^{2} - 12y + 9\), identify the coefficients as follows:

• \(a = 4\), the coefficient of \(y^{2}\)
• \(b = -12\), the coefficient of \(y\)
• \(c = 9\), the constant term

Recognizing these coefficients allows you to assign values to \(a, b,\) and \(c\) to aid in the process of factoring the quadratic equation. Identifying them correctly is crucial for determining whether the expression fits into a special form like a perfect square trinomial.

Factoring Methods

Factoring methods are techniques used to break down complex quadratic expressions into simpler parts or products of simpler expressions. Factoring can be especially straightforward with a perfect square trinomial like \(4y^{2} - 12y + 9\).

To factor a perfect square trinomial, write it in the form of \(a^{2} - 2ab + b^{2}\). In this instance:

• Identify \(a\) as \(2y\)
• Identify \(b\) as \(3\)
• Other steps include verifying \(-2ab\) matches the middle term \(-12y\).

With these parameters confirmed, factor the whole expression as \((2y - 3)^{2}\).

Remember, practice and familiarity with different factoring methods will simplify recognizing patterns leading to efficient factoring of any quadratic equation.